T II E 



AMERICAN HOUSE-CARPENTER 



A TEEATISE 



ON 



THE ART OF BUILDING, 



AND 



THE STRENGTH OF MATERIALS. 




K. G. HATFIELD, Architect, 



MEM. AM, mST. OF ARCHITECTS. 



SEVENTH EDITION, REVISED AND ENLARGEfl 

WITH ADniTIONAL ILLUSTRATIONS. 



NEW YORK: 
JOHN WILEY & SON, 



15 ASTOB PLACE. 

1871 







"Entered according to Act of Congi-ess, in the year 186T, by 

E. G. HATFIELD, 

la th« Clerk's Office of the District Court of the United States, for the Southern District 
of New York. 






« 



< 



J H\ F. Trow & So^ 
2C3-^ij East lauti St.XNj^ York, 



WV'' 



Tg 



s^; 



PREP^ACE. 



This book is intended for carpenters — for masters, 
journeymen and apprentices. It lias long been the com- 
plaint of tbis class that architectural books, intended for 
their instruction, are of a price so high as to be placed 
beyond their reach. This is owing, in a great measure, 
to the costliness of the plates with which they are ilhis- 
trated; an unnecessary expense, as illustrations upon 
wood, printed on good paper, answer every useful pur- 
pose. Wood engravings, too, can be distributed among 
the letter-press ; an advantage which plates but partially 
possess, and one of great importance to the reader. 

Considerations of this kind induced the author to 
undertrke the preparation of this volume. The subject 
matter has been gleaned from works of the first autho- 
rity, and subjected to the most careful examination. 
The explanations have all been written out from the 
figures themselves, and not taken from any other work ; 
and the figures have all been drawn exj)ressly for this 
book. In doing this, the utmost care has been taken to 
make everything as plain as the nature of the case 
would admit. 

The attention of the reader is particularly directed to 
the following new inventions, viz ; an easy method of 
describing the curves of mouldings through three gi\eD 



I V PREFACE. 

points ; a rule to determine the projection of eave cor 
nices ; a new method of proportionnig a cornice to a 
lai-ger given one ; a way to determine the lengths and 
bevils of rafters for hip-roofs ; a way to proportion the 
rise to the tread in stairs ; to determine the true position 
of butt-joints in hand-rails ; to find the bevils for splayed- 
work ; a general rule for scrolls, &c. Many problems in 
geometry, also, have been simplified, and new ones intro- 
duced. Much labour has been bestowed upon the sec- 
tion on stairs, in which the subject of hand-railing is 
presented, in many respects, in a new, and it is hoped, 
more practical form than in previous treatises on that 
subject. 

The author has endeavoured to present a fund of use- 
ful information to the American house-carpenter that 
would enable him to excel in his vocation ; how far he 
has been successful in that object, the book itself must 
determine. 

New York, Oct. 15, 1844. 



FIFTH EDITION. 



Since the first edition of this work was published, 1 
have received numerous testimonials of its excellent 
practical value, from the very best sources, viz. from the 
workmen themselves who have used it, and avIio have 
profited by it. As a convenieMt manual for reference in 
respect to every question rela :ng either to the simplei 
operations of Carp'^.ntry ^r the more intricate and 



PTiEFACE. 



abstruse problems of Geometry, those wlio have ti iecl 
it assure me that they have been gi-eatly assisted in using 
it. And, indeed, to the true workman, there is, in the 
study of the subjects of which this volume treats, a con- 
tinual source of profitable and pleasurable interest. 
Gentlemen, in numerous instances, have placed it in the 
hands of their sons, who have manifested a taste for 
practical studies ; and have also procured it for the nse 
of the workmen ujDon their estates, as a guide in their 
mechanical operations. I was not, then, mistaken in my 
impressions, that a work of this kind was wanted ; and 
this evidence of its usefulness rewards me in a measure 
for the pains taken in its preparation. 

New York, Oct. 1, 1852. 



SEYEISTTII EDITIOK 

It is now thirteen years since the first edition of tht 
American House Carpenter was published. The attempt 
to furnish the recipients of this book a,- ith a fund of 
useful information in a compact and accessible form, has 
been so far successful that the sixth edition was exhausted 
nearly a year ago. At that time it was determined, 
before issuing another edition, to make a thorough 
revision of the work. The time occupied in this labour 
has been unexpectedly prolonged by at least six months, 
and this ha,s resulted from various causes, but more 
especially from the absorbing nature of my professional 
duties. A large portion of the work has been rewiitten. 



VI . PREFACE. 

aLout 130 pages of new matter iriti duced and manj 
new cuts inserted. 

Tlie most important additions to the work will be 
found in tlie section on Framing or Construction. Here 
will be found, now first published, the results of experi- 
ments on such building materials as are in common use 
in this country, and an extended series of rules for the 
application of this experimental knowledge to the 2:)rac- 
tical purposes of building. Some of the rules are new, 
while others heretofore in use have been simplified. 
This section has been much improved, and it is hoped 
that it will be of service, not only to the house carpenter 
but also to the architect and civil engineer. 

In preparing the original work, a desire to state the 
subjects treated of in terms suited to the comprehension 
of all classes of workmen, precluded the use of algebra- 
ical symbols and formulae. In this edition, hov/ever, it 
has been deemed best to introduce them wherever they 
would contribute to the clearer elucidation of the sub- 
ject; but care has been taken to state them in a simple 
form at first, and so to explain the symbols as they are 
introduced that those heretofore uninstructed in regard 
to them, may comprehend what little is here exhibited, 
and at the same time be induced to pursue the study more 
fully in works more strictly mathematical. But for those 
who may not succeed in comprehending the algebraical 
formulae, it may be stated that all the practical deductions 
derived from them are written out in words at length, 
80 as to be fully understood without their assistance. 

E. G. H. 

New York, ,^epi. 1. 186T. 



TABLE OF COXTENTS. 



rKTBODUCTiON. — Directions for Drawing 



1-14 



SECTION I.— PRACTICAL GEOMETRY 



Definitions 

Problems on Lines and A ngles 
Problems on the Circle .... 

Problems on Polygons .... 

Problems on Proportions . . , • 
Problems on the Conic Sections . . , 
Demonstrations. — Definitions, Axioms, &c. . 
Demonstrations. — Propositions and Corollaries 



15-70 

71-80 
81-92 
93-106 
107-110 
111-128 
130-l;}9 
140- I G7 



SECTION II.— ARCHITECTURE. 



History 

Styles. — Origin, Definitions, Proportions 
Grecian Orders. — Doric, Ionic and Corinthian 
Roman Orders. — Doric, Ionic, Corinthian and Composite 
Egyptian Style . . . . 

BuUdhigs generally 

Plans and Elevation for a City Dwelling 
Principles of Architecture. — Requisites in a Building 
r'rinoiples of Construction. — The Foundations, Column 
Principles of Construction. — The Wall, Lintel, Arch 
Principles of Construction. — The Vault, Dome, Roof 



168-181 
182-106 
197-211 
212-215 
216, 217 
218-223 
223, 224 
225-229 
230-232 
233-235 
23«i-23S 



J/ipjui&Z- 



H- 



- llO 



VUl TABLE OF CONTENTS. 



SECTION III.— MOULDINGS, C0KNICE8, &o. 

AEn 
Mouldings.— Elements, Examples . . . . . 239-25C 

Cornices. — Designs ■ . .~ . 251 

Cornices. — Problems 252-256 



SECTION IV.— FEAMING, OE CONSTEUCTION. 

First Principles. — Laws of Pressure ....... 257-282 

Resistance of Materials. — Strength, Stiffness 283-2S(i 

Resistance to Compression. — Various kinds 287-290 

Results of Experiments on American Materials, Tables I., 11. . . 291-293 

Practical Rules and Examples 294-305 

Resistance to Tension 306 

Results of Experiments on American Materials, Table III. . . . 307, 308 

Practical Rules and Examples 309-316 

Resistance to Cross Strains. — Strength, Stiffness 317-319 

Resistance to Deflection. — Stillness, Formulae 320-322 

Practical Rules and Examples ........ 323-326 

Table lY.— Weight on Beams, Formulae 326 

Practical Rules and Examples 327-329 

Table Y. — Dimensions of Beams, Formulae 329 

Resistance to Rupture — Strength 331 

Results of Experiments on A.merican Materials, Table VL ... 331 

Table YIL— Safe Weight on Beams, Formulae ...... 333 

Practical Rules and Examples 333 

Table YIII, — Dimensions of Beams, Formulae . • . . . , 334 

Practical Rules and Examples 334 

Systems of Framing, Simplicity of Designs 33.3 

Floors. — Yarious, Cross-furring, Reduction of Formulae , . . 336, 337 

Practical Rules and Examples 338-344 

Bridging-strips, Girders, Precautions . . . . . . . 34.'i-349 

Pariiiiom. — Examples, Load on Partitions, &c 350-353 

Roofs. — Stability, Inclination 354, 355 

Load. — Roofing, Truss, Ceiling, Wind, Snow . • ... 356-358 

Strains. — Yertical, Oblique, Horizontal . . ... 359-368 

Resistancs of the Material in Rafter and Tie-beam .... 368 

Dimensions. — Rafter, Braces, Tie-beam, Iron Roda .... 370-37-1 

Practical Rules and E^xamples ' . 375-383 

Table IX.— Weight of Roofs, per Foot 376 



TABLE OF CONTENTS. 



IX 











ARTa, 


Examples of Roofs 384-cs£8 


Problems for Hip-rafter ... 






387-388 


Domes. — Examples Area of Ribs 






. 389-391 


Problems in Domes .... 








392-398 


Bridges. — Examples .... 






> 


399-401 


Rules for Dimensions .... 






. 


401-405 


Abutments and Piers .... 






. 


406, 407 


Stone Bridges, Centreing 






. 


408-417 


Joints in Timberwork .... 








. 418-427 


^on Work.— Pins, Nails, Bolts, Straps 








428 


Iron-Girflers. — Cast Girder, Bow-string, Brick Arch 




. 429-435 


Practical Rules and Examples 








431-435 



SECTION v.— DOORS, WINDOWS, &a 



Doors. — Dimensions, Proportions, Examples 
"Windows. — Form, Size, Arrangement, Problems 



436-441 
442-448 



SECTION VI.— STAIES. 



Principles, Pitch Board 

Platform Stairs, Cylinders, Rail, Face Mould 
Winding Stairs, Falling Mould, Face Mould, Joints 
Elucidation of Butt Joint ..... 
Quarter-circle Stairs. — Falling Mould, Pace Mould 
Face Mould. — Elucidation .... 
Face Moulds. — Applied to Plank, Bevils, &c. 
Face Moulds. — Another method .... 
Scrolls, Rule, Falling and Face Moulds, Xewel Cap 



449-456 

457-463 
469-476 

477 
478-480 

481 
482-484 
485-188 
489-498 



SECTION VI1.-SI1AD0W8. 



Shadows on Mouldings, Curves, Inclinations, &c. 
Shadows. — Reflected Light .... 



623 



APPENDIX 



Algebraical Signs 
rrigouomotrical Terms 



3 



X T^BLE OF CONTENTS. 

PA6B 

Glossary of Architectural Terms . 7 

Tables of Squares, Cubes and Roots .18 

Rules for Reduction of Decimals 27 

Table of Areas and Circumferences of Circlea .... 29 

Table of Capacity of Wells, Cisterns, Ac. 33 

Table of Areas of Polygons, &c. ..*••• . 34 

Table of Weights of Materials • • .IS 



INTRODUCTION. 



Art. 1. — A knowledge of the properties and principles of lines 
can best be acquired by practice. Although the various problems 
throughout this work may be understood by inspection, yet they 
will be impressed upon the mind with much greater force, if they 
are actually performed with pencil and paper by the student. 
Science is acquired by study — art by practice : he. therefore, who 
would have any thing more than a theoretical, (which must of 
necessity be a superficial,) knowledge of Carpentry, will attend 
to the following directions, provide himself with the articles here 
specified, and perform all the operations described in the follow- 
ing pages. Many of the problems may appear, at the first read- 
ing, somewhat confused and intricate ; but by making one line 
at a time, according to the explanations, the student will not 
only succeed in copying the figures correctly, but by ordinary 
attention will learn the principles upon which they are based, 
and thus be able to make them available in any unexpected case 
to which they may apply. 

2. — The following articles are necessary for drawing, viz : a 
drawing-board, paper, drawing-pins or mouth-glue, a sponge, a 
T-square, a set-square, two straight-edges, or flat rulers, a lead 
pencil, a piece of india-rubber, a cake of india-ink, a set of draw- 
ing-instruments, and a scale of equal parts. 

3. — The size of the drawing-hoard must be regulated accord- 

qr to the size of the drawings which are to be made upon it. 
Yet for ordinary practice, in learning to draw, a board about 15 

1 



i5 AMERICAN HOUSE CARPENTER. 

by 20 inches, and one inch thick, will be found large enough, 
and more convenient than a larger one. This board should be 
well-seasoned, perfectly square at the corners, and without 
clamps on the ends. A board is better without clamps, because 
the little service they are supposed to render by preventing the 
board from warping, is overbalanced by the consideration that 
the shrinking of the panel leaves the ends of the clamps project- 
ing beyond the edge of the board, and thus interfering with the 
proper working of the stock of the T-square. When the stufl 
is well-seasoned, the warping of the board will be but trifling ; 
and by exposing the rounding side to the fire, or to the sun, it 
may be brought back to its proper shape. 

4. — For mere line drawings, it is unnecessary to use tbe hest 
drawing-paper ; and since, where much is used the expense will 
be considerable, it is desirable for economy to procure paper 
of as low a price as will be suitable for the purpose. The best 
paper is made in England and marked "Whatman." This is 
a hand-made paper. There is also a machine-made paper at 
about half-price, and the Manilla paper, of various tints of rus- 
set color, is still less in price. These papers are of the various 
sizes needed, and are quite sufficient for ordinary drawings. 

5. — A drawing-pin is a small brass button, having a steel pin 
projecting from the under side. By having one of these at each 
corner, the paper can be fixed to the board ; but this can be done 
in a much better manner with mouth-glue. The pins will pre- 
vent the paper from changing its position on the board ; but, 
more than this, the glue keeps the paper perfectly tight and 
smooth, thus making it so much the more pleasant to work on. 

To attach the paper with mouth-glue, lay it with the bottom 
side up, on the board ; and with a straight-edge and penknife, 
cut off the rough and uneven edge. With a sponge moderately 
wet, rub all the surface of the paper, except a strip around the 
edge about half an inch wide. As soon as the glistening of the 
water disappears, turn the sheet over, and place it upon tJie 



INTRODUCTION. 6 

board just where you wish it ghied. Commence upon one of 
the longest sides, and proceed thus : lay a flat ruler upon the 
paper, parallel to the edge, and within a quarter of an inch of it 
With a knife, or any thing similar, turn up the edge of the papei 
against the edge of the ruler, and put one end of the cake oi 
mouth-glue between your lips to dampen it. Then holdmg it 
upright, rub it against and along the entire edge of the paper 
that is turned up against the ruler, bearing moderately against 
the edge of the ruler, which must be held firmly with the left 
hand. Moisten the glue as often as it becomes dry, until a 
sufficiency of it is rubbed on the edge of the paper. Take 
away the ruler, restore the turned-up edge to the level of the 
board, and lay upon it a strip of pretty stiff paper. By rubbing 
upon this, not very hard but pretty rapidly, with the thumb nail 
of the right hand, so as to cause a gentle friction, and heat to be 
imparted to the glue that is on the edge of the paper, you will 
make it adhere to the board. The other edges in succession 
must be treated in the same manner. 

Some short distances along one or more of the edges, may 
afterwards be found loose : if so, the glue must again be applied, 
and the paper rubbed until it adheres. The board must then be 
laid aAvay in a warm or dry place ; and in a short time, the sur- 
face of the paper will be drawn out, perfectly tight and smooth, 
and ready for use. The paper dries best when the board is laid 
level. When the drawing is finished, lay a straight-edge upon 
the paper, and cut it from the board, leaving the glued strip still 
attached. This may afterwards be taken off by wetting it freely 
with the sponge ; which will soak the glue, and loosen the 
paper. Do this as soon as the drawing is taken off, in order that 
the board may be dry when it is wanted for use again. Care 
must be taken that, in applying the glue, the edge of the paper 
does not become damper than the rest : if it should, the paper 
must be laid aside to dry, (to use at another time,) and anotheJ 
sheet be used in its place. 



4 AMERICAN HOUSE CARPENTER. 

Sometimes, especially when the drawing board is new, the 
paper will not stick very readily ; but by persevering, this diffi- 
culty may be overcome. In the place of the mouth-glue, a 
strong solution of gum-arabic may be used, and on some 
accounts is to be preferred ; for the edges of the paper need not 
be kept dry, and it adheres more readily. Dissolve the gum in 
a sufficiency of warm water to make it of the consistency of 
linseed oil. It must be applied to the paper with a brush, when 
the edge is turned up against the ruler, as was described for the 
mouth-glue. If two drawing-boards are used, one may be in use 
while the other is laid away to dry ; and as they may be cheaply 
made, it is advisable to have two. The drawing-board having 
a frame around it, commonly called a panel-board, may afford 
rather more facility in attaching the paper when this is of the 
size to suit ; yet it has objections which overbalance that con 
sideration. 

6 — A T-square of mahogany, at once simple in its construc- 
tion, and affording all necessary service, may be thus made, 
Let the stock or handle be seven inches long, two and a quarter 
inches wide, and three-eighths of an inch thick: the blade, 
twenty inches long, (exclusive of the stock,) two inches wide, 
and one-eighth of an mch thick. In joining the blade to the 
stock, a very firm and simple joint may be made by dovetailing 
it — as shown at Fig. 1. 




Fig.1. 



INTRODUCTION. 

7. — The set-square is in the form of a right-angled triangle ; 
and is commonly made of mahogany, one-eighth of an inch in 
thickness. The size that is most convenient for general use, is 
six inches and three inches respectively for the sides which con- 
tain the right angle ; although a particular length for the sides is 
by no means necessary. Care should be taken to have the square 
corner exactly true. This, as also the T-square and rulers^ 
should have a hole bored through them, by which to hang them 
upon a nail when not in use. 

8. — One of the rulers may be about twenty inches long, and 
the other six inches. The pencil ought to be hard enough to 
retain a fine point, and yet not so hard as to leave ineffaceable 
marks. It should be used lightly, so that the extra marks that 
are not needed when the drawing is inked, may be easily rubbed 
off with the rubber. The best kind of india-ink is that which 
will easily rub off upon the plate ; and, when the cake is rub- 
bed against the teeth, will be free from grit. 

9. — The drawing-instruments may be purchased of mathe- 
matical instrument makers at various prices : from one to one 
hundred dollars a set. In choosing a set, remember that the 
lowest price articles are not always the cheapest. A set, com- 
prising a sufficient number of instruments for ordinary use, well 
made and fitted in a mahogany box, may be purchased of the 
mathematical instrument-makers in New York for four or live 
dollars. But for permanent use those which come at ten or 
twelve dollars will be found to be the best. 

10.— The best scale of equal parts for carpenters' use, is one 
that has one-eighth, three-sixteenths, one-fourth, three-eighths, 
one-half, five-eighths, three-fourths, and seven-eighths of an 
inch, and one inch, severally divided into twelfths^ instead ot 
being divided, as they usually are, into tenths. By this, if it be 
required to proportion a drawing so that every foot of the object 
represented will upon the paper measure one-fourth of an inch, 
use that part of the scale which is divided into one-fourths of an 



6 AMERICAN HOUSE-CARPENTER. 

inch^ taking for every foot one of those divisions, and for every 
inch one of the subdivisions into twelfths ; and proceed in like 
manner in proportioning a drawing to any of the other divisions 
of the scale. An instrument in the form of a semi-circle, called a 
protractor^ and used for laying down and measuring angles, is 
of much service to surveyors, but not much to carpenters. 

11. — In drawing parallel lines, when they are to be parallel 
to either side of the board, use the T-square ; but when it is 
required to draw lines parallel to a line which is drawn in a 
direction oblique to either side of the board, the set-square must 
be used. Let a 6, {Fis^. 2,) be a line, parallel to which it is 




Fig % 



desired to draw one or more lines. Place any edge, as c c?, ol 
the set-square even with said, line ; then place the ruler, g h, 
against one of the other sides, as c e, and hold it firmly ; slide 
the set-square along the edge of the ruler as far as it is desired, 
as at/; and a line drawn by the edge, i/, will be parallel to a b. 
12.— To draw a line, as k Z, {Fig. 3,) perpendicular to another, 
as a b, set the shortest edge of the set-square at the line, ab; 
place the ruler against the longest side, (the h3rpothenuse of the 
right-angled triangle ;) hold the ruler firmly, and slide the set- 
square along until the side, e d touches the point, k ; then the 
line, I k, drawn by it, will be perpendicular to a b. In like 



INTRODUCTION. 



manner the drawing of other problems may be facilitated, as will 
be discovered in using the instruments. 




Fig. 3. 



13. — ^In drawing a problem, proceed, with the pencil sharpened 
to a point, to lay down the several lines until the whole figure is 
completed; observing to let the lines cross each other at the 
several angles, instead of merely meeting. By this, the length 
of every line will be clearly defined. With a drop or two of 
vater, rub one end of the cake of ink upon a plate or saucer, 
until a suiRciency adheres to it. Be careful to dry the cake of 
ink ; because if it is left wet, it will crack and crumble in pieces. 
With an inferior camel's-hair pencil, add a little water to the 
ink that was rubbed on the plate, and mix it well. It should be 
diluted sufficiently to flow freely from the pen, and yet be thick 
enough to make a Mack line. With the hair pencil, place a 
little of the ink between the nibs of the drawing-pen, and screw 
the nibs together until the pen makes a fine line. Beginning 
with the curved lines, proceed to ink all the lines of the figure ; 
being careful now to make every line of its requisite length. II 
they are a trifle too short or too long, the drawing will have a 
ragged appearance; and this is opposed to that neatness and 
accuracy which is indispensable to a good drawing. When the 
ink is dry, efface the pencil-marks with the india-rubber. If 



8 AMERICAN HOUSE-CARPENTER. 

the pencil is used lightly, they will all rub off, leaving those 
lines only that were inked. 

14. — In problems, all auxiliary lines are drawn light ; while 
the lines given and those sought, in order to be distinguished at 
a glance, are made much heavier. The heavy lines are made 
so, by passing over them a second time, having the nibs of the 
pen separated far enough to make the lines as heavy as desired. 
If the heavy lines are made before the drawing is cleaned with 
the rubber, they will not appear so black and neat ; because the 
india-rubber takes away part of the ink. If the drawing is a 
ground-plan or elevation of a house, the shade-lines, as they are 
termed, should not be put in until the drawing is shaded; as 
thel-e is danger of the heavy lines spreading, when the brushy in 
shading or coloring, passes over them. If the lines are inked 
with common writing-ink, they will, however fine they may be 
made, be subject to the same evil ; for which reason, india-ink 
is the only kind to be used. 



THE 



AMERICAN HOUSE-CARPENTER. 



SECTION L— PRACTICAL GEOMETRY. 



DEFINITIONS. 



15. — Geometry treats of the properties of magnitudes, 

16. — A point has neither length, breadth, nor thickness. 

17. —A line has length only. 

18. — Superficies has length and breadth only. 

19. — A plane is a surface, perfectly straight and even in every 
direction ; as the face of a panel when not warped nor winding. 

20. — A solid has length, breadth and thickness. 

21. — A right, or straight^ line is the shortest that can be 
drawn between two points. 

22. — Parallel lines are equi-distant throughout their length. 

23. — An angle is the inclination of two lines towards orw^ 
anoth(ir. {Fig. 4.) 




\ 



Fig. 4. 



Fig. 5. 

2 



Fig. & 



10 



AMERICAXr HOUSE-CARPZNTER. 



24. — A right angle has one line perpendicular to the othei. 
{Fig. 5.) 

25. — An oblique angle is either greater or less than a right 
angle. {Fig- 4 and 6.) 

26. — An acute angle is less than a right angle. {Fig. 4.) 

27. — An obtuse angle is greater than a right angle. [Fig. 6.) 

When an angle is denoted by three letters, the middle one, in 
the order they stand, denotes the angular point, and the other 
two the sides containing the angle ; thus, let « 6 c, {Fig, 4,) be 
the angle, then b will be the angular point, and a b and b c will 
be the two sides containing that angle. 

28. — A triangle is a superficies having three sides and angles, 

[Fig. 7, 8, 9 and 10.) 





Fig. 7. 



Fig. 8. 



29. — An equi-lateral triangle has its three sides equal. 
[Fig. 7.) 

30. — An isosceles triangle has only two sides equal. {Fig. 8.) 
31. — A scalene triangle has all its sides unequal. {Fig. 9) 





Fig. lU. 



32.— A right-angled triangle has one right angle. {Fig. 10.) 

33. — An acute-angled triangle has all its angles acute. 
{Fig. 7 and 8.) 

34. — An obtuse-angled triangle has one obtuse angle. 
{Fig. 9.) 

35.— A quadrangle has four sides and four angles. {Fig. 11 
to 16.) 



PRACTICAL GEOMETRY. 



n 




Fig. 11. 



Fi^. 12. 



36. — A parallelogram is a quadrangle having its opposite 
sides parallel. {Fig. 11 to 14.) 

37. — ^A rectangle is a parallelogram, its angles being righi 
angles. [Fig. 11 and 12.) 

38. — A square is a rectangle having equal sides. {Fig. 11 .) 

39. — A rhombus is an equi-lateral parallelogram having ob- 
lique angles. {Fig. 13.) 




7 



Fig. 13. 



Fig. 14. 



40. — A rhomhoid is a parallelogram having oblique angles. 
(Fig. 14.) 

41. — A trapezoid is a quadrangle having only two of its sides 
parallel. {Fig. 15.) 




Fig. 15. 



Fig. 16. 



42. — A trapezium is a quadrangle which has no two of its 
sides parallel. {Fig. 16.) 

43. — A polygon is a figure bounded by right lines. 

44. — A regular polygon has its sides and angles equal. 

45. — An irregular polygon has its sides and angles unequal. 

46.— A trigon is a polygon of three sides, {Fig. 7 to 10 ,) 
a tetragon has four sides, {Fig. 11 to 16 ;) a pentagon has 



12 



AMERICAN HOUSE-CARPENTER. 



five, {Fig. 17 ;) a hexago7i six, {Fig. 18 ;) a heptagon seven, 
[Fig.- 19;) an octagon eight, [Fig. 20;) Si nonago7i nine; a 
decagon ten ; an undecagon eleven ; and a dodecas^on twelve 
sides. 






Fig. 17. 



Fig. 18, 



Fig. 19. 



Fig. 20. 



47. — A circle is a figure bounded by a curved line, called the 
circumference ; which is every where equi-distant from a cer- 
tain point within, called its centre. 

The circumference is also called the j^eriphery^ and sometimes 
the circle. 

48. — The radius of a circle is a right line drawn from the 

centre to any point in the circumference, [a b, Fig. 21.) 

All the radii of a circle are equal. 



/ 

c 


^ \ 


"^ 


^\l 






a 




\ 


c 




J 


-.f- 


v_ 


^^ 


/ 

i- 



Pig. 21. 



49. — The diameter is a right line passing through the centre, 
and terminating at two opposite points in the circumference. 
Hence it is twice the length of the radius, (c c?. Fig. 21.) 

50. — An arc of a circle is a part of the circumference, (c h or 
hed, Fig. 21.) 

51. — A chord is a right line joining the extremities of an arc. 



PRACTICAL GEOMETRY. 



13 



52. — A segment is any part of a circle bounded by an arc and 
its chord. {A, Fig. 21.) 

53. — A sector is any part ^f a circle bounded by an arc and 
two radii, drawn to its extremities. {B^ Fig. 21.) 

54. — A quadrant^ or quarter of a circle, is a sector having a 
quarter of the circumference for its arc. (C, Fig. 21.) 

55. — A tangent is a right line, which in passing a curve, 
touches, without cutting it. (/^, Fig. 21.) 

56. — A cone is a solid figure standing upon a circular base 
diminishing in straight lines to a point at the top, called it^ 
vortex. {Fig. 22.) 




Fig. 22. 




57. — The axis of a cone is a right line passnig through it, froir. 
the vertex to the centre of the circle at the base. 

58. — An ellipsis is described if a cone be cut by a plane, not 
parallel to its base, passing quite through the curved surface, 
(a 6, Fig. 23.) 

59. — A jparahola is described if a cone be cut by a plane, 
parallel to a plane touching the curved surface, (c c?. Fig. 23 — 
c d being parallel ^ofg.) 

60. — An hyperbola is described if a cone be cut by a plane^ 
parallel to any plane within the cone that passes through its 
vertex, (e A, Fig. 23.) 

61. — Foci are the points at which the pins are placed in de- 
scribing an ellipse. (See Art. 115, and/, /, Fig. 24.) 



14 



AMERICAN HOUSE-CARPENTER. 




62. — The transverse axis is the longest diameter" of the 
ellipsis, [a b, Fig. 24.) 

63. — The co7ijngate axis is the shortest diameter of the 
ellipsis ; and is, therefore, at right angles to the transverse axis, 
(c d, Fig. 24) 

64. — The parameter is a right line passing through the focus 
of an ellipsis, at right angles to the transverse axis, and termina- 
ted by the curve, {g h and g t^ Fig. 24.) 

65. — A diameter of an ellipsis is any right line passing 
through the centre, and terminated by the curve, [k /, or m w, 
Fig. 24.) 

66. — A diameter is conjugate to another when it is parallel to 
a tangent drawn at the extremity of that other — thus, the diame- 
ter, m ?z, {Fig. 24,) being parallel to the tangent, o p, is therefore 
conjugate to the diameter, k I. 

67. — A double ordinate is any right line, crossing a diameter 
of an ellipsis, and drawn parallel to a tangent at the extremity of 
that diameter. (^ ^, Fig. 24.) 

68. — K cylinder is a solid generated by the revolution of a 
right-angled parallelogram, or rectangle, about one of its sides ; 
and consequently the ends of the cylinder are equal circles. 
[Fig. 2o.) 



PRACTICAL GEOMETRY. 



15 



Fig, 25. 




Fig. 26. 



69. — The axis of a cylinder is a right line passing through it, 
from the centres of the two circles which form the ends. 

70. — A segment of a cylinder is comprehended under three 
planes, and the curved surface of the cylinder. Two of these 
are segments of circles : the other plane is a parallelogram, called 
by way of distinction, the ]jlane of the segment. The circular 
segments are called, the ends of the cylinder. {Fig. 26.) 



i\^. B. — For Algebraical Signs, Trigonometrical Terms, &c., 
see Appendix. 



PROBLEMS 



RfGHT LINES AND ANGLES. 



71. — j?b bisect a line. Upon the ^.ndf of the line, a b, {Fig. 
27,) as centres, with any distance for radius greater than halt 




Fig. 27 



ek 6j describe arcs cutting each other in c and d ; draw the line, 
c d, and the point, e, where it cuts a b, will be the middle of the 
line, a b. 

In practice, a line is generally divided with the compasses, or 
dividers ; but this problem is useful where it is desired to draw, 
at the middle of another line, one at right angles to it. (See 
Art. 85.) 

d 

4^ 




Fig. 28. 



72, — To erect a perpendicular. From the point, a, (Fig. 28,) 



PRACTICAL GEOMETRY. 



17 



set oft any distance, as a b, and the same distance from a to c ^ 
upon c, as a centre, with any distance for radius greater than c a, 
describe an arc at d ; upon 6, with the same radius, describe 
another at d ; join d and a, and the hne, d a, will be the per- 
pendicular required. 

This, and the three following problems, are more easily per- 
formed by the use of the set-square — (see Art. 12.) Yet they 
are useful when the operation is so large that a set-square cannot 
be used. 




Fif . 29. 

73. — To let fall a 'perpeiidicidar. Let a, [Fig. 29,) be the 
point, above the line, h c, from which the perpendicular is re- 
quired to fall. Upon a, with any radius greater than a d^ de- 
scribe an arc, cutting h c dX e and/; upon the points, e and/ 
with any radius greater than e d, describe arcs, cutting each 
other at g- ; join a and g, and the line, a d, will be the perpen- 
dicular required. 




Fig. 30. 

74. — To erect a perpendicular at the end of a line. Let a, 
{Fig. 30,) at the end of the line, c a, be the point at which the 
perpendicular is to be erected. Take any point, as 6, above the 
line, c a, and with the radius, h a, describe the arc, d a e. 
through d and 6, draw the line, d e ; join e and a, then e a will 
be the perpendicular required 



18 



AMERICAN HOUSE-CARPENTER. 



The principle here made use of, is a very important one ; and 
is applied in many other cases — (see Art. 81, b. and Art. 84. 
For proof of its correctness, see Ai^t. 156.) 




74, a. — A second method. Let b, {Fig. 3] ,) at the end of the 
Iin«^, a b, be the point at which it is required to erect a perpen- 
dicular. Upon b, with any radius less than b a, describe the arc, 
c e d ; upon c, with the same radius, describe the small arc at e, 
and upon e, another at d ; upon e and d, with the same or any 
other radius greater than half e c?, describe arcs intersecting at/; 
join/ and &, and the line,/&, will be the perpendicular required. 
This method of erecting a perpendicular and that of the fol- 
lowing article, depend for accuracy upon the fact that the side 
of a hexagon is equal to the radius of the circumscribing circle. 




d 

Fig. 32. 



74, b. — A third method. Let 6, {Fig. 32,) be the given point 
at which it is required to erect a perpendicular. Upon b^ with any 
radius less than b a, describe the quadrant, d ef; upon c?, with 
the same radius, describe an arc at e, and upon c, another at c , 



PRACTICAL GEOMETRY. 19 

throigh d and e, draw d c, cutting the arc in c; jcin c and 6, 
then c h will be the perpendicular required. 

This problem can be solved by the six^ eight and ten rule 
as it is called ; which is founded upon the same principle as 
the problems at Art. 103, 104; and is applied as follows. Let 
a d^ [Fig. 30,) equal eight, and a e, six ; then, \i d e equals ten, 
the angle, e a c?, is a right angle. Because the square of six 
and that of eight, added together, equal the square of ten, thus : 
6 X 6 === 36, and 8 X 8 = 64 ; 36 + 64 = 100, and 10 x 10 = 
100. Any sizes, taken in the same proportion, as six, eight and 
ten, will produce the same effect : as 3, 4 and 5, or 12, 16 and 
20. (See Art. 103.) 

By the process shown at Fig. 30, the end of a boc^d may be 
squared without a carpenters'-square. All that is necessary is a 
pair of compasses and a ruler. Let c a be the edge of the board, 
and a the point at which it is required to be squared. Take the 
point, 6, as near as possible at an angle of forty-five degrees, or on 
a mitre-line, from a, and at about the middle of the board. This 
is not necessary to the working of the problem, nor does it affect 
its accuracy, but the result is more easily obtained. Stretch the 
compasses from b to a, and then bring the leg at a around to d ; 
draw a line from d, through 6, out indefinitely ; take the dis- 
tance, d b, and place it from b to e ; join e and a ; then e a will 
be at right angles to c a. In squaring the foundation of a build- 
ing, or laying-out a garden, a rod and chalk-line may be used 
instead of compasses and ruler. 

7^. — To let fall a perpendicular near the end of a line. 

Let e, [Fig. 30,) be the point above the line, c a, from which the 

perpendicular is required to fall. From e, draw any line, as e d, 

obliquely to the line, c a ; bisect e d at b ; upon 6, with the 

radius, b e, describe the arc, e a d ; join e and a ; then e a will 

be the perpendicular required. 




76. — To make an angle., (as e df Fig. 33,) equal to a given 
angle, (as /> a c.) From the angular point, a, with any radius 
describe the arc, b c ; and with the same radius, on the line, d e, 



20 



AMERICAN HOUSE-CARPENTER. 



and from the point, c?, describe the arc^fg; take the distance^ 

6 c, and upon g, describe the small arc at/; join /and d ; and 

the angle, e df, will be equal to the angle, b a c. 

If the given line upon which the angle is to be made, is situa- 
ted parallel to the similar line of the given angle, this may be 
performed more readily with the set-square. (See Art. 11.) 




Fig. 34. 



77. — To bisect an angle. Let a b c, [Fig. 34,) be the angle 

to be bisected. Upon 6, with any radius, describe the arc, a c ; 

upon a and c, with a radius greater than half a c, describe arcs 

cutting each other at d ; join b and d ; and b d will bisect the 

angle, a b c, as was required. 

This problem is frequently made use of in solving other pro- 
blems ; it should therefore be well impressed upon the memory. 




Fig. 35. 

78. — To trisect a right ang'e. Upon «, {Fig. 35,) with any 

radius, describe the arc, b c ] upiui 5 and c, with the same radius, 

describe arcs cutting the arc, 6 c, at c? and e ; from d and e, draw 

lines to a, and they will trisect the angle as was required. 

The truth of this is made evident by the following operation. 
Divide a circle into quadrants : also, take the radius in the divi- 
ders, and space off the circumference. This will divide the 
circumference into just six parts. A semi-circumference, there- 



PRACTICAL GEOMETRY. 21 

tore, IS equal to three, and a quadrant to one and a half of those 
parts. The radius, therefore, is equal to f of a quadrant ; and 
this is equal to a right angle. 



'i 



Fifr. 36. 

79. — Through a given point, to draw a line parallel to a 
given line. Let «, {Fig. 36,) be the given point, and h c the 
given line. Upon any point, as d, in the line, h c, with the 
radius, d a, describe the arc, a c; upon a, with the same radius, 
describe the arc, d e ; make d e equal to a c ; through e and a 
draw the line, e a ; which will be the line required. 

This is upon the same principle as Art. 76. 




HO. — To divide a given line into any number of equal parts. 

Let a bj {Fig. 37,) be the given line, and 5 the number of parts. 

Draw a c, at any angle to a -6 ; on a c, from a, set off 5 equal 

parts of any length, as at 1, 2, 3, 4 and c ; join c and b ; through 

the points, 1, 2, 3 and 4, draw 1 e, 2/, 3 ^ and 4 A, parallel tc 

c b ; which will divide the line, a b, as was required. 

The lines, a b and a c, are divided in the same proportion. 
(See Art. 109.) 

T'dE CIRCLE. 

81. — To find the centre of a circle. Draw any chord, as a i6 



^2 



AMERICAN HOUSE-CARPENTER. 




[Fig. 38,) and bisect it Avith the perpendicular, c d ; lisect c a 
with the line, ef, as at g- ; then g is the centre as was required. 




81, a. — A second method. Upon any two points in the cir- 
cumference nearly opposite, as a and h. {Fig. 39,) describe arcs 
cutting each other at c and d : take any other two |»oints, as e 
and/, and describe arcs intersecting as at g and h ; join^ and A, 
and c and d ; the intersection, o, is the centre. 

This is upon the same principle as Art. 85. 




81, b. — A third method. Draw any chord, as a h. [Fig. 40,) 



PRACTICAL GEOMETRY. 2^^ 

and from the point, a, draw a c, at right angles to a b ; join 

c and b ; bisect c 6 at €> — which will be the centre of the circle. 

If a circle be not too large for the purpose, its centre may very 
readily be ascertained by the help of a carpenters'-square, thus : 
app' y the corner of the square to any point in the circumference, 
as at a ; by the edges of the square, (which the lines, a b and 
a c, represent,) draw lines cutting the circle, as at b and c ; join 
b and t ; then if 6 c is bisected, as at d, the point, d, will be the 
centre. (See Art. 156.) 




fig. 41. 



82. — At a given poi?it in a circle, to draw a tangent thereto. 
Let a, {Fig. 41,) be the given pomt, and b the centre of the cir- 
cle. Join a and b ; through the point, a, and at right angles to 
a 6, draw c d ; then cdis the tangent required. 




Fig. 42 

S3. — The same, without making use of the centre of the 
circle. Let a, {Fig. 42,) be the given point. From a, set oi? 
any distance to 6, and the same from b to c ; join a and c , 
upon a, with a b for radius, describe the arc, d b e ; make d b 
equal to be; through a and d, draw a line ; this will be the 
tangent required. 

The correctness of this method depends upon the fact that 
the angle formed by a chord and tangent is equal to any 



24 



AMERICAN HOUSE-CAliFENTER. 



inscribed angle in the opposite segment of tlie circle, {Art, 
163 f) ah being the chord, and lea the angle in the opposite 
segment of the circle. 'Now, the angles d ah and h c a are 
equal, because the angles d ah and h a e are, by construction, 
equal ; and the angles h a c and h c a are equal, because the 
triangle ah c is an isosceles triangle, having its two sides, a h 
and h c, bj construction equal ; therefore the angles d ah and 
h c a are equal. 




Fiff. 43. 



84. — A circle a/nd a tangent given, to find the jpoint of con- 
tact. From any point, as a, (Fig. 43,) in the tangent, h c, draw 
a line to the centre d / bisect a d at e ; upon e, with the radius, 
e a, describe the arc, af d ; f is the point of contact required. 

If/* and d were joined, the line would form right angles with 
the tangent, h c. (See Art. 156.) 




Fig. 44. 



85. — Through any three jpoints not in' a st/raight line, to drma 
a circle. Let a, h and c, {Fig. 44,) be the three given points. 
Upon a and h, with any radius greater than half a h, describe 



PKACTICAL GEOMETRY. 



25 



arcs intersecting a / d and e / npon h and c, witn an j radius 
greater than half h c, describe arcs intersecting at / and g ', 
through d and 6, draw a right line, also another through f and 
g / upon the intersection, ^, with the radius, A a^ describe the 
circle, ah c^ and it will be the one required. 




Fig. 45. 



86. — Three joints not in a st/raight line heing given, tojmd 

a fourth that shall, with the three, lie in the circumferenGe of a 

circle. Let ate, {Fig. 45,) be the given points. Connect 

them with right lines, forming the triangle, a ch ; bisect the 

angle, ch a, {Art. 77,) with the line h d ; also bisect c a in e, 

and erect e d, perpendicular to a c, cutting h din d ; then d is 

\hQ fourth point required. 

A fifth point may be found, as at^, by assuming a, d and I, 
as the three given points, and proceeding as before. So, also, 
any number of points may be found ; simply by using any three 
already found. This problem will be serviceable in obtaining 
short pieces of very flat sweeps. (See Art. 397.) 

The proof of the correctness of this method is found in the 

fact that equal chords subtend equal angles, {Art. 162.) Join 

d and c ; then since a e and e c are, by construction, equal, 

therefore the chords a d and d c are equal ; hence the angles 

they subtend, d h a and dh c, are equal. So likewise chords 

drawn from a tof and from y to d, are equal, and subtend the 

equal angles, dhfsmdfh a. Additional points, heyond a or 

1), may be obtained on the same principle. To obtain a jDoint 

beyond a, on h, as a centre, describe with any radius the arc 

i n, make o n equal to o i ; through h and n draw h g ; on a as 



26 



AMERICAN HOUSE-CAKPENTER. 



centre and with afioT radius, describe tlie arc, cutting ^ & at 
^5 then g is the point sought. 




87. — To describe a segment of a circle hy a set-triangle. Let 
a 5, {Fig. 46,) be the chord, and c d the height of the segment. 
Secure two straight-edges, or rulers, in the position, c e and cf 
by nailing them together at ^, and affixing a brace from e to 
// put in pins at a and 5/ move the angular point, c, in the 
direction, a oh j keeping the edges of the triangle hard against 
the pins, a and 5 / a pencil held at c wdll describe the arc, a ch. 

A curve described by this process is accurately circulanr^ and 

is not a mere approximation to a circular arc, as some may 

suppose. This method produces a circular curve, because all 

inscribed angles on one side of a chord line are equal. {Art. 

161.) To obtain the radius from a chord and its versed sine. 

see Art. 165. 

If the angle formed by the rulers at c be a right angle, the 
segment described w^ill be a semi-circle. This problem is use* 
ful in describing centres for brick arches, when they are re- 
quired to be rather flat. Also, for the head hanging-stile of a 
window-frame, where a brick arch, instead of a stone lintel, is 
to be placed over it. 

87 a. — To find the radius of an arc of a circle when the 

chord and versed sine are given. The radius is equal to the 

sum of the squares of half the chord and of the versed sine, 

divided by twice the versed sine. This is expressed, algebraic- 



ally, thus — r= 



(i)M-y 

2?; 



, where r is the radius, c the chord, and v 



the versed sine. {Art. 165.) 

Example. — In a given arc of a circle, a chord of 12 feet ha? 



PEACTICAL GEOMETEY. 



27 



tne rise at the middle, or the versed sme, equal to 2 feet, what 

is the radius ? 

Half the chord equals 6, the square of 6 is, 6 x 6 = 36 
The square of the versed sine is, 2 x 2 == 4 

Their sum equals, 40 

Twice the versed sine equals 4, and 40 divided by 4 equals 10 
Therefore the radius, in this case, is 10 feet. This result in 
&hown in less space and more neatly by using the above alge- 
braical formula. For the letters, substituting their value, tlie 

formula r =: -^^-P^ — becomes r = ^^ — — -, and performing 

^2 V 2x2 ^ ^ 

the arithmetical operations here indicated equals 
6^ + 2^ _ 36 + 4 __ 40 _ 

~~i ~ "T - 4 ~ 

87 b. — To find the versed sine of an arc of a circle when the 

radius and chord are given. The versed sine is equal to the 

radius, less the square root of the difference of the squares of 

the radius and half chord : expressed algebraically thus — v = r 

— \^ r^ — (|)^ where r is the radius, v the versed sine, and e 

the chord. {Art. 158.) 

Mcample. — In an arc of a circle whose radius is 75 feet, 
what is the versed sine to a chord of 120 feet ? By the table 
in the Appendix it will be seen that — 

The square of the radius, 75, equals, . . 5625 
Tlie square of half the chord, 60, equals, . 3600 



Tlie difference is, 2025 

The square root of this is, . . . . 45 
This deducted from the radius, . . . 75 

The remainder is the versed sine, — 30 

This is expressed by the formula thus — 
V =. 75 - V7¥'-ljf'Y=15 - V 5625 - 3600 == 75 — 45 = 3C 



1 h J 




28 



AMERICAN HOTJSE-CAICPENTEE. 



88. — To describe the segment of a circle hy inter sect ton of 

lines. Let a 5, {Fig. 47,) be the chord, and c d the height of 

the segment. Through c, draw e f parallel to ah ; draw h f 

at right angles to ch ; make c e equal to of; draw a g and 

h h^ at right angles to ah ; divide c e^ cf d a^ dh^ a g^ and 

h A, each into a like number of equal parts, as four ; draw the 

lines, 1 1, 2 2, &c., and from the points, (9, o and <?, draw lines 

to G / at the intersection of these lines, trace the curve, a ch^ 

which will be the segment required. 

In very large work, or in laying out ornamented gardens, 
&c., this will be found useful ; and where the centre of the 
proposed arc of a circle is inaccessible it will be invaluable. 
(To trace the curve, see note at Art. 117.) 

The lines e a^ c d and f ^, would, were they extended, meet 

in a point, and that point would be in the opposite side of the 

circumference of the circle of which a ch h a segment. The 

lines 1 1, 2 2, 3 3, would likewise, if extended, meet in the 

same point. The line, c d^ if extended to the opposite side of 

the circle, would become a diameter. The line,/* J, forms, by 

construction, a right angle with h c, and hence the extension of 

f h would also form a right angle with h c, on the opposite side 

oi h G ; and this right angle would be the inscribed angle in 

the semicircle ; and since this is required to be a right angle, 

{Art. 156,) therefore the construction thus far is correct, and it 

will be found likewise that at each point in the curve formed 

by the intersection of the radiating lines, these intersecting 

lines are at right angles. 




88 a. — Points in the circumference of a circle may be ob- 
tained arithmetically, and positively accurate, by the calcula- 
tion of ordinates^ or the parallel lines, 1, 2, 3, 4. (Fig 



PEACTIOAL GEOACETET. 29 

4:7 a) These orditiates are drawn at right angles to the chord 
line, a 5, and they may be drawn at any distance apart, eithei 
equally distant or unequally, and there may be as many of 
them as is desirable ; the more there are the more points in 
the curve wdll be obtained. If they are located in pairs, 
equally distant from the versed sine, c d^ calculation need be 
made only for those on one side of c d^ as those on the opposite 
side will be of equal lengths, respectively ; for example, 1, on 
the left-hand side of c d^ is equal to 1 on the right-hand side, 
2 on the right equals 2 on the left, and in like manner for 
the others. 

The length of any ordinate is equal to the square root of 
the difference of the squares of the radius and abscissa, less 
the difference between the radius and versed sine. {Art. 166.) 
The abscissa being the distance from the foot of the versed sine 



to the foot of the ordinate. Algebraically, y — yjf— x^ — 
{r — ^'), where y is put to represent the ordinate ; a?, the ab- 
scissa ; ^', the versed sine ; and r^ the radius. 

Example. — An arc of a circle has its chord, a 5, {Fig. 47 a^ 
100 feet long, and its versed sine, c <^, 6 feet. It is required to 
ascertain the length of ordinates for a sufficient number of 
points through wdiich to describe the curve. To this end it is 
requisite, first, to ascertain the radius. This is readily done in 

accordance with Art S7 a. For, -^^^^- , becomes — 



2v ' 2 X 5 

252*5 = radius. Having the radius, the curve might at once 
be described without the ordinate points, but for the impracti- 
cability that usually occurs, in large, flat segments of the circle, 
of getting a location for the centre ; the centre usually being 
inaccessible. Tlie ordinates are, therefore, to be calculated. 
In Fig. 47 a the ordinates are located equidistant, and are 10 
feet apart. It will only be requisite, therefore, to calculate 
those on one side of the versed sine, c d. For the first ordinate, 
1, the formula, y = yf r' — x^ — {r — v) becomes 
y ~ ^252-5'^ - IJf - (2 52-5 - 5). 

= V6375M5 - 100 - 247*5. 

= 252*3019 - 247-5. 

= 4-8019 = the first ordinate, 1. 



30 



AMloEICA^ HOUSB -CAKPENTEE. 



For 


tne becond — • 






For 


y = v'^5^'5'' - 
= 251-7066 
=z 4-2066 

the third — 


20' -(252-5 - 5). 

- 247-5. 

= the second ordinate, 02. 


y = V252-5-^ - 
= 250-7115 
=z 3-2115 
For the fourth — 


30=^ _ 247-5. 
-- 247-5. 
= the third 


ordinate, 03. 



y = -v/252-5' - 40' - 247*5. 

= 249-3115 - 247-5. 

= 1-8115 = the fourth ordinate, 04. 

The results here obtained are in feet and decimals of a foot. 
To reduce these to feet, inches, and eighths of an inch, proceed 
as at Reduction of Decimals in the Appendix. If the two-feet 
rule, used by carpenters and others, were decimally divided, 
there would be no necessity of this reduction, and it is to be 
hoped that the rule will yet be thus divided, as such a reform 
would much lessen the labor of computations, and insure more 
accurate measurements. 



Yersed sine, 


c d, = 


ft. 


. 5-0 


n: 


ft. 


, 5-0 inches. 


Ordinates, 


01,^ 




4-8019 


— 




4-9|- inches nearly. 


a . 


02,= 




4-2066 


= 




4"22 inches nearly. 


u 


3,^ 




3-2115 


= 




3-2^ inches nearly. 


a 


04,= 




1-8115 


= 




1-91 inches nearly. 




Fig. 48. 



89. — In a given angle, to describe a tanged curve. Let ah c, 

{Fig. 48,) be the given angle, and 1 in the line, (^ 5, and 5 in 

the line, h c, the termination of the curve. Divide 1 h and h 5 

into a like number of equal parts, as at 1, 2, 3, 4 and 5 ; join 1 

and 1, 2 and 2, 3 and 3," &c. ; and a regular curve will be 

formed that will be tangical to the line, a 5, at the. point, 1, and 

to h G Bi 5. 

This is of much use ir. stair-building, in easing the angles 
formed between the wall-string and the base of the hall, also 



PRACTICAL GEOMETRY. 



31 



between the front string and level facia, and in many other 
instances. The curve is not circular, but of the form of the 
parabola, {Fig. 93 ;) yet in large angles the difference is not 
perceptible. This problem can be applied to describing the 



Fig. 49. 

curve for door heads, window-heads, &c., to rather better ad- 
vantage than Art. 87. For instance, let a J, {Fig. 49,) be the 
width of the opening, and g d the height of the arc. Extend g 
d^ and make d e equal to c d ; join a and e^ also e and h / and 
proceed as directed above. 




?>ff. 50 



90. — To descrihe a circle within any given triangle, so that 
the sides of the triangle shall he tangical. Let ah c^ {Fig. 50,) 
be the given triangle. Bisect the angles a and 5, according to 
Art. 77 ; upon ^, the point of intersection of the bisecting lines, 
with the rudius, d e, describe the required circle. 




Fig :,i. 



32 



AI^IEEICAN HOUSE-CAEPENTEE. 



91. — About a given circle^ to describe an eqid-lateral tri- 
angle. Let a db G^ {Fig. 51,) be tlie given circle. Draw the 
diameter, c d ; upon d^ with the radius of the given circle, 
describe the arc, aeb ; join a and b ; draw/* ^, at right angles 
to d G ; make/* c and c g^ each equal to ab ; from f^ through 
a^ drsiwfh, also from g, through 5, draw g h; then/ g h will 
be the triangle required. 




92. — To jhid a right line nearly equal to the circumferenee 
of a circle. Let ab c d^ {Fig. 52,) be the given circle. Draw 
the diameter, a g ; on this erect an equi-lateral triangle, a e c^ 
according to Art. 93 ; draw g /*, parallel to a g ; extend e c to 
y, also e a to g ; then g f will be nearly the length of the 
semi-circle, a d g ; and twice g f will nearly equal the circum- 
ference of the circle, ab g d^2i^ was required. 

Lines drawn from ^, through any points in the circle, as <?, o 
and {?, to jp^]) and p, will divide g f in the same way as the 
semi-circle, a d c^i^ divided. So, any portion of a circle may 
be transferred to a straight line. This is a very useful pro- 
blem, and should be well studied ; as it is frequently used to 
solve problems on stairs, domes, &c. 

92, a. — Another method. Let a bf c, {Fig. 53,) be the given 

circle. Draw the diameter, ac j from d, the centre, and at right 

angles to a <?, draw db i join b and g j bisect b g at e ; from d^ 

through e, draw dfi then ^y* added to three times the diameter, 

will equal the circumference of the circle sufficieutly near for 



PRACTICAL (JEOMETET. 33 




Fig. 53. 



many uses. The result is a trifle too large, If tlie circumfer- 
erence found by this rule, be divided by 64:8*22, the quotient 
will be the excess. Deduct this excess, and the remainder 
will be the true circumference. This problem is rather more 
curious than useful, as it is less labor to perform the operation 
arithmetically: simply multiplying the given diameter by 
3*1416, or where a great degree of accuracy is needed by 
8-1415926. 

POLYGONS, &0. 




93. — ZPpon a given line to const/ruct an equi-ldteral oriangve. 
Let a 5, {Fig. 54,) be the given line. Upon a and J, with a h 
for radius, describe arcs, intersecting at c ; join a and c, also c 
and 1) / then a oh will be the triangle required. 

94. — To describe an egui-lateral rectangle^ or sqiia/re. Let 
a 5, {Fig, 55,) be the length of a side of the pi^oposed square. 
Upon a and 5, with a h for radius, describe the arcs a d and 
he; bisect the arc, a e, in // upon e, with e f for radius, de- 

5 



S4 



AMEEICAN HOUSE-CAIII'ENTEE. 
d 




Fig. 55, 



scribe the arc, cfdj join a and c, g and d^ d and Z> / fhen a e 
d h will be the sqnare required. 




95. — Within a given cirde^ to inscrihe cm equidateral tri- 
angle^ hexagon or dodecagon. Let ah g d^ {Fig. 56,) be the 
given circle. Draw the diameter, h d ; upon J, with the 
radius of the given circle, describe the arc, a e g j join a and 
c, also a and d^ and c and d — and the triangle is completed. 
For the hexagon: from a^ also from c, through ^, draw the 
lines, af and eg; join <^ and 5, 5 and c, g and /*, &c., and the 
hexagon is completed. The dodecagon may be formed by 
bisecting the sides of the hexagon. 

Each side of a regular hexagon is exactly equal to the, 
radius of the circle that circumscribes the figure. For the 
radius is equal to a chord of an arc of 60 degrees ; and, as 
every circle is supposed to be divided into 360 degrees, there 
is just 6 times 60, or 6 arcs of 60 degrees, in the whole circum- 
ference. A line drawn from each angle of the hexagon to the 
centre, (as in the figure,) divides it into six equal, equi-lateral 
triangles. 

96. — Within a square to im^scfnbe om> octagon. Let ah cd^ 



PEAOTICAL GEOMETET. 



35 




{Fig. 57,) be tlie given square. Draw the diagonals, a d and 
he; upon a^ 5, g and d^ with a e for radius, describe arcs cut- 
ting the sides of the square at 1, 2, 3, 4, 5, 6, 7 and 8 ; join 1 
and 2, 3 and 4, 5 and 6, &c., and the figure is completed. 

In order to eight-square a hand-rail, or any piece that is to 
be afterwards rounded, draw the diagonals, a d and h c, upon 
the end of it, after it has been squared-up. Set a gauge to the 
distance, a «, and run it upon the whole length of the stuff, 
from each corner both ways. This will show how much is to 
be chamfered off, in order to make the piece octagonal. {Art, 
159.) 




Fig. 58. 



Fig. 59, 



97. — Within a given circle to inscribe any regular polygo7i. 
Let ah g2, {Fig. 58, 59 and 60,) be given circles. Draw the 
diameter, a c; upon this, erect an equi-lateral triangle, a e c, 
according to Art. 93 ; divide a c into as many equal parts as 
the polygon is to have sides, as at 1, 2, 3, 4, &c. ; from ^, 



36 



AMERICAiq- HOTISE-CARPENTEE. 



through each even number, as 2, 4, 6, &c., draw Hnes cutting 
the circle in the points, 2, 4, &c. ; from these points and at 
right angles to a c, draw lines to the opposite part of the circle * 
this will give the remaining points for the polygon, as 5,/, &c. 
In forming a hexagon, the sides of the triangle erected upon 
c, (as at Fig. 59,) mark the points l and /. This method of 
locating the angles of a polygon is an approximation suffi- 
ciently near for many purposes ; it is based upon the like prin- 
ciple with the method of obtaining a right line nearly equal to 
a circle. {Art. 92.) The method shown at Art. 98 is accurate. 



a 




Fig- 61. 



Fig. 62. 



Fig. 63. 



98. — Upon a given line to describe cmy regular polygon. 

Let a 5, (Fig. 61, 62 and 63,) be given lines, equal to a side of 

the required figure. From 5, draw h c, at right angles to ah ; 

upon a and 5, with a h for radius, describe the arcs, a g d and 

f eh I divide a c into as many equal parts as the polygon is to 

have sides, and extend those divisions from c towards d; from 

the second point of division counting from c towards a, as 3, 

{Fig. 61,) 4, {Fig. 62,) and 5, {Fig. 63,) draw a line to h; take 

the distance from said point of division to a, and set it from h 

to e; join e and a ; upon the intersection, o, with the radius, 

a, describe the circle a f d h ; then radiating lines, drawn 

from h through the even numbers on the arc, a d^ will cut the 

circle at the several angles of the required figure. 

In the hexagon, {Fig. 62,) the divisions on the arc, a d^ are not 
necessary ; for the point, 6>, is at the intersection of the arcs, a d 
andy Z>, the points, /" and d^ are determined by the intersection of 
those arcs with the circle, and the points above, g and A, can be 
found by drawing lines from a and 5, through the centre, o. In 
polygons of a greater number of sides than the hexagon, the in- 
tersection, (?, comes above the arcs ; in such case, therefore, the 



PRACTICAL GEOMETEY. 



37 



lines a e and h 5, [Fig. 63,) have to be extended before they will 
intersect. This method of describing polygons is founded on 
correct principles, and is therefore accurate. In the circle equal 
arcs subtend equal angles, [Arts. 86 and 162.) Although this 
method is accurate, yet polygons may be described as accu- 
rately and more simply in the following manner. It will be 
observed that much of the process in this metho.d is for the pur 
pose of ascertaining the centre of a circle that will circumscribe 
the proposed polygon. By reference to the Table of Polygons 
in the Appendix it will be seen how this centre may be obtained 
arithmetically. This is the Rule. — Multiply the given side by 
the tabular radius for polygons of a like number of sides with 
the proposed figure, and the product will be the radius of the 
required circumscribing circle. Divide this circle into as many 
equal parts as the polygon is to have sides, connect the points of 
division by straight lines, and the figure is complete. For exam- 
ple : It is desired to describe a polygon of 7 sides, and 2Cf inches 
a side. The tabular radius is 1*1523824:. This multiplied by 
20, the product, 23*04:7648 is the required radius in inches. The 
Rules for the Reduction of Decimals, also in the Appendix, 
show how to change decimals to the fractions of a foot or an 
inch. ■ From this, 23*047648 is equal to 23tV inches nearly. It 
is not needed to take all the decimals in the table, three or four of 
them will give a result sufiiciently near for aU ordinary practice. 



f 




Fig. 64. 

99. — To consifiruct a triangle whose sides shall he severally 
equal to three given lines. Let a, h and <?, {Fig. 64,) be the given 
lines. Draw the line, d e^ and make it equal too / upon e^ with 
h for radius, describe an arc at /*/ upon d^ with a for radius, 
describe an arc intersecting the other at/*/ join d andy, also 
f and e / then df e will be the triangle required. 




Fig. 65. 



Fig. 66. 



38 



PRACTICAL GEOMETRY. 



100. — To construct a figure equal to a g\V6n^ right-lined 

figure. Let abed, {Fig. 65,) be the given figure. Make ef, 

{Fig. 66,) equal to c d ; upon /, with d a for radius, describe an 

arc at g; upon e, with c a for radius, describe an arc intersecting 

the other at g ; join g and e ; upon / and g, with d b and a b 

for radius, describe arcs intersecting at h ; join g and A, also h 

and/; then Fig. 66 will everyway equal Fig. 65. 

So, right-lined figures of any number of sides may be copied, 
oy first dividing them into triangles, and then proceeding as 
above. The shape of the floor of any room, or of any piece of 
land, (fee, may be accurately laid out by this problem, at a scale 
upon paper ; and the contents in square feet be ascertained by 
the next. 



e X 



Fig. 67. 

101. — To make a parallelogram, equal to a given triangle. 

Let a 6 c, {Fig. 67,) be the given triangle. From a, draw a d, 

at right angles to be; bisect a d in e ; through e, draw/^, 

parallel to b c ; from b and c, draw b f and c g, parallel to d e ; 

then b f g c will be a parallelogram containing a surface exactly 

equal to that of the triangle, a b c. 

Unless the parallelogram is required to be a rectangle, the lines, 
b f and c g, need not be drawn parallel to d e. If a rhomboid is 
desired, they may be drawn at an oblique angle, provided they 
be parallel to one another. To ascertain the area of a triangle, 
multiply the base, b c, by half the perpendicular height, d a. In 
doing this, it matters not which side is taken for base. 



A ^ 

^^ e 
.y^ C 



Fig. 68. 



/ 



AMERICAN HOUSE-CARPENTER. 



39 



102. — A parallelogram being given, to construct anothei 
equal to it, and having a side equal to a given line. Let A 
[Fig. 68,) be the given parallelogram, and B the given line 
Produce the sides of the parallelogram, as at a, b, c and d ; make 
e d e:iual to B ; through d, draw c /, parallel to g b ; through 
c, draw the diagonal, c a ; from a, draw a /, parallel to e d 
then C will be equal to A. (See Art. 144.) 



A 


a ^^**^^ 




B 



Fig 69. 

103. — To make a square equal to two or more given squares . 
Let A and B, [Fig. 69,) be two given squares. Place them so 
ds to form a right angle, as at a ; join b and c ; then the square, 
C, formed upon the line, b c, will be equal in extent to the squares, 
A and B, added together. Again : if a b, {Fig. 70,) be equal to 




the side of a given square, c a, placed at right angles to a b, be the 
side of another given square, and c d, placed at right angles to 



40 



PRACTICAL GEOMETRY. 



c b, be the side of a third given square ; then the square, A^ 
formed upon the Hne, d b, will be equal to the three given 
squares. (See Art. 157.) 

The usefulness and importance of this problem are proverbial. 
To ascertain the length of braces and of rafters in framing, the 
length of stair-strings, &c., are some of the purposes to which it 
may be applied in carpentry. (See note to Art. 74, b.) If the 
length of any two sides of a right-angled triangle is known, that 
of the third can be ascertained. Because the square of the 
hypothenuse is equal to the united squares of the two sides that 
contain the right angle. 

(1.) — The two sides containing the right angle being known, 
to find the hypothenuse. Rule. — Square each given side, add 
the squares together, and from the product extract the square- 
root : this will be the answer. For instance, suppose it were 
required to find the length of a rafter for a house, 34 feet wide, — 
the ridge of the roof to be 9 feet high, above the level of the 
wall-plates. Then 17 feet, half of the span, is one, and 9 feet, 
the height, is the other of the sides that contain the right angle 
Proceed as directed by the rule : 



17 
17 



119 
17 



81 = square of 9. 
289 = square of 17. 



289 = square of 17. 370 Product. 



.Q^ . 1 ) 370 ( 19-235 + 
1 1 



29 ) 270 
9 261 



square-root of 370 ; equal 19 feet, 2^ in. 
nearly : which would be the required 
length of the rafter. 



382)- -900 
2 764 



3843 ) 13600 
3 11529 



38465)* 207100 (By reference to the table of square-roots 
192325 in the Appendix, the root of almost any 

number may be found ready calculated ; 

also, to change the decimals of a foot to inches and parts, see 
Eules for the Keduction of Decimals in the Appendix.) 



AMERICAN HOUSE-CARPENTER. 4] 

Agair : suppose it be required, in a frame building, to rind the 
length of a brace, having a run of three feet each way from the 
point of the right angle. The length of the sides containing the 
right angle will be each 3 feet : then, as before— 

3 
3 

9 = square of one side. 
3 times 3 = 9 = square of the other side. 

1 8 Product : the square-root of which is 4*2426 -f ft., 
or 4 feet, 2 inches and |ths. full. 

(2.) — The hypothenuse and one side being known, to find the 
other cide. Rule. — Subtract the square of the given side from 
the square of the hypothenuse, and the square-root of the product 
will be the answer. Suppose it were required to ascertain the 
greatest perpendicular height a roof of a given span may have, 
when pieces of timber of a given length are to be used as rafters. 
Let the span be 20 feet, and the rafters of 3 X 4 hemlock joist. 
These come about 13 feet long. The known hypothenuse, 
then, is 13 feet, and the known side, 10 feet — that being half the 
span of the building. 

13 
13 



39 
13 



169 = square of hypothenuse- 
lO times 10 = 100 = square of the given side. 



69 Product : the square-root of which is 8 
•3066 + feet, or 8 feet, 3 inches and ^ths. full. This will be 
the greatest perpendicular height, as required. Again : suppose 
that in a story of 8 feet, from floor to floor, a step-ladder is re- 
quired, the strings of which are to be of plank, 12 feet long ; and 
it is desirable to know the greatest run such a length of string 
will afford. In this case, the two given sides are — hypothenuse 
12, perpendicular 8 feet. 

12 times 12 = 144 = square of hypothenuse. 
8 times 8 = 64 = square of perpendicular. 



80 Product : the square-root of which is 8-9442 -f- 
feet, or 8 feet, ""1 inches and i^gths. — the answer, as required. 

6 



42 



PRACTICAL GEOMETRY. 



Many other cases might be adduced to show the utility of this 
problem. A practical and ready method of ascertaining the 
length of braces, rafters, (fcc, when not of a great length, is to 
apply a rule across the carpenters'-square. Suppose, for the 
length of a rafter, the base be 12 feet and the height 7. Apply 
the rule diagonally on the square, so 'that it touches 12 inches 
from the corner on one side, and 7 inches from the .corner on the 
other. The number of inches on the rule, which are intercepted 
by the sides of the square, 13^ nearly, will be the length of the 
rafter in feet ; viz, 13 feet and |ths of a foot. If the dimensions 
are large, as 30 feet and 20, take the half of each on the sides of 
the square, viz, 15 and 10 inches ; then the length in inches 
across, will be one-half the number of feet the rafter is long. 
This method is just as accurate as the preceding ; but when 
the length of a very long rafter is sought, it requires great care 
and precision to ascertain the fractions. For the least variation 
on the square, or in the length taken on the rule, would make 
perhaps several inches difference in the length of the rafter. 
For shorter dimensions, however, the result will be true enough. 




Fig. 71. 



104. — To make a circle equal to two given circles. Let ^ 
and i?, [Fig. 71,) be the given circles. In the right-angled tri- 
angle, a h Cj make a h equal to the diameter of the circle, B^ and 
c b equal to the diameter of the circle, A ; then the hypothenuse^ 




Pig. 78. 



AMERICAN HOUSE-CARPENTER. 



43 



a c, will be the diameter of a circle, C, which will be equal in 

area to the two circles, A and B^ added together. 

Any polygonal figure, as A, {Fig. 72,) formed on the hypo- 
thenuse of a right-angled triangle, will be equal to two similar 
figures,* as B and C, formed on the two legs of the triangle. 



Fig. 73 

105. — To construct a square equal to a given rectangle, 
Let A, [Fig. 73,) be the given rectangle. Extend, the side, a 6, 
and make h c equal to 6 e ; bisect a c in/, and upon /, with the 
radius, / a, describe the semi-circle, age; extend e b, till it 
cuts the curve in g ; then a square, b g h d, formed on the line, 
b ^, wall be equal in area to the rectangle, A, 



e 

b 

A 



c g 

Fig. 74. 

105, a. — Another method. Let A, {Fig. 74,) be the given 
rectangle. Extend the side, a b, and make a d equal to a c , 

* Similar figures are such as have their several angles respectively equal, and theij 
aidjs respectively proportionate. 



44 PRACTICAL GEOMETRY. 

biseiit a d in e ; upon e, with the radius, e a, describe the semi 
circle, a f d ; extend^ h till it cuts the curve in/; join a and 
/'; then the square, B, fprmed on the line, a/, will be equal in 
area to the rectangle, A. (See Art. 156 and 157.) 

106. — To form a square equal to a given triangle. Let a 6, 
[Fig. 73,) equal the base of the given triangle, and h e equal 
half its perpendicular height, (see Fig. 67 ;) then proceed a? 
directed at Art, 105. 




107. — Two right lines being giveji, to find a third propor- 
tional thereto. Let A and -B, {Fig. 7b^) be the given lines. 
Make a h equal to A ; from a, draw a c, at any angle with a b ; 
make a c and a d each equal to B ; join c and b ; from d, draw 
d e, parallel to c b ; then a e will be the third proportional re- 
quired. That is, a e bears the same proportion to B, as B does 
to A. 




iOS.— Three right lines being given, to find a fourth pro 
portional thereto. Let A, B and C, {Fig. 76,) be the given 
lines. Make a b equal to A ; from a, draw a c, at any angle 
with a b; make a c equal to B, and a e equal to C ; join c and 
b ; from e, draw ef parallel io c b ; then a f will be the fourth 
proportional required. That is, a f bears the same proportioi^ 
to C, as B does to A, 



AMERICAN HOUSE-CARPENTER. 



45 



To apply this problem, suppose the two axes of a given ellipsis 
and the longer axis of a proposed ellipsis are given. Then, by 
this pfoblem, the length of the shorter axis to the proposed ellip- 
sis, can be found ; so that it will bear the same proportion to the 
longer axis, as the shorter of the given ellipsis does to its longer. 
(See also, Art. 126.) 



A 

P 


















c 




^ 


-^ 


\ 


\\\ 


a 


1 


Fig 


3 

77. 


4 5 6 



109. — A line with certain divisions being giveii, to divide 

another, longer or shorter, given line in the same proportion. 

Let A, {Fig. 77,) be the line to be divided, and B the line with 

its divisions. Make a b equal to B, with all its divisions, as at 

1,2, 3, &c. ; from a, draw a c, at any angle with a b ; mike a c 

equal to A ; join c and b ; from the points, 1, 2, 3, &c., draw 

lines, parallel to c 6 ; then these will divide the line, a c, in the 

same proportion as B is divided — as was required. 

This problem will be found useful in proportioning the mem- 
bers of a proposed cornice, in the same proportion as those of a 
given cornice of another size. (See Art. 253 and 254.) So of 
a pilaster, architrave, &c. 




110. — Betioeen two given right lines^ to find a mean pi o* 
portional. Let A and 5, {Fig. 78,) be the given lines. On 
the line, a c, make a b equal to A, and b c equal to B ; bisect a 
c in e ; u}»on e, with e a for radius, describe the semi -circle, a d 



46 



AMERICAN HOUSE-CARPENTER. 



c ; at 5, erect I d^ at right angles to a c ; then I d will be the 
mean proportional between A and B. That is, alhioh d ^% 
1) d i^ioh G, This is nsaally stated thus — a h *» h b v,h d l h c^ 
and since the product of the means equals the product of the 
extremes, therefore, al xl G = h ct. This is shown geometri- 
cally at Art. 105. 

CONIC SECTIONS. 

111. — li a cone, standing upon a base that is at right angles 
with its axis, be cut by a plane, perpendicular to its base and 
passing through its axis, the section will be an isosceles triangle ; 




Fiff. 79. 



\BiS> al)G^ Fig. 79 ;) and the base will be a semi-circle. If a cone 
be cut by a plane in the direction, e /*, the section will be an 
eW&psis ; if in the direction, m Z, the section will be Sijpardbola ; 
and if in the direction, r c, an hyperbola. (See Art. 56 to 60.) K 
the cutting planes be at right angles with the plane, abc^ then — 
112. — To find the axes of the ellipsis^ bisect ef^ {Fig. 79,) in 
g 'j through ^, draw h % parallel to ab ; bisect h i inj'y upon 
j, with j h for radius, describe the semi-circle, hJci^ from g^ 
draw g ^, at right angles to hi; then twice g h wiD be the 
conjugate axis, and e/* the transverse. 



AMERICAN HOUSE-CARPENTER. 



47 



113. — Tofijid the axis and base of the parabola. Let m /, 
[Fig. 79,) parallel to a c, be the direction of the cutting plane 
From m^ draw m c?, at right angles to a b ; then I m will be the 
axis and height, and m c? an ordinate and half the base ; as at 
Fig. 92, 93. 

114. — To find the height^ base and transverse axis of an 
hyperbola. Let o r, [Fig. 79,) be the direction of the cutting 
plane. Extend o r and a c till they meet at n ; from o, draw 
p, at right angles to a b; then r o will be the height, n r the 
transverse axis, and o p half the base ; as at Fig. 94. 




Fig. 80. 



115. — The axes being given, to find the foci, and to describe 

mi ellipsis with a string. Let a b, {Fig. 80,) and c d, be the 

given axes. Upon c, with a e or 6 e for radius, describe the arc, 

ff; then /and/, the points at which the arc cuts the transverse 

axis, will be the foci. At/ and /place two pins, and another at c ; 

tie a string about the three pins, so as to form the triangle, // c ; 

remove the pin from c, and place a pencil in its stead ; keeping the 

string taut, move the pencil in the direction, eg a; it will then 

describe the required ellipsis. The lines, /^ and g f show the 

position of the string when the pencil arrives at g. 

This method, when performed correctly, is perfectly accurate; 
but the string is liable to stretch, and is, therefore, not so good to 
Lise as the trammel. In making an ellipse by a string or twine, 
that kind should be used which has the least tendency to elasticity, 
For this reason, a cotton cord, such as chalk-lines are commonly 
made of, is not proper for the purpose : a linen, or flaxen cord is 
mu3h better. 



4S PRACTICAL GEOMETRY. 




Fig. 81 



116. — The axes being give?!, to describe an ellipsis with a 
trammel. Let a 6 and c d, {Fig. 81,) be the given axes. Place 
the trammel so that a line passing through the centre of the 
grooves, would coincide with the axes ; make the distance from 
the pencil, e, to the nut,/, equal to half c d ; also, from the pen- 
cil, e, to the nut, ^, equal to half a b ; letting the pins under the 
nuts slide in the grooves, move the trammel, e g, in the direction, 
c b d ; then the pencil at e will describe the required ellipse. 

A trammel maybe constructed thus : take two straight strips ot 
board, and make a groove on their face, in the centre of their 
width ; join them together, in the middle of their length, at right 
angles to one another ; as is seen at Fig. 81. A rod is then to be 
prepared, having two moveable nuts made of wood, with a mor- 
tice through tliem of the size of the rod, and pins under them 
large enough to fill the grooves. Make a hole at one end of the 
rod, in which to place a pencil. In the absence of a regular tram- 
mel, a temporary one may be made, which, for any short job, 
will answer every purpose. Fasten two straight-edges at right 
angles to one another. Lay them so as to coincide with the axes 
of the proposed ellipse, having the angular point at the centre. 
Then, in a rod having a hole for the pencil at one end, place two 
brad-awls at the distances described at Art. 116. While the 
pencil is moved in the direction of the curve, keep the brad-awls 
hard against the straight-edges, as directed for using the tram- 
mel-rod, and one-quarter of the ellipse will be drawn. Then 
by shifting the straight-edges, the other three quarters in succes- 
sion may be drawn. If the required ellipse be not too large, a 
carpenter s'-squ are may be made use of, in place of the straight- 
edges. 

An improved method of constructing the trammel, is as fol 
lows : make the sides of the grooves bevilling from the face ot 
the stuff, or dove-tailing instead of square. Prepare two slips ot 
wood, each about two inches long, which shall be of a shape to 
•'ust fill the groove when slipped in at the end. These, instead ot 



AMERICAN HOUSE-CARPFNTER. 



49 



pins, are to be attached one to each of the moveable nuts with 
a screw, loose enough for the nut to move freely about the screw 
as an axis. The advantage of this contrivance is, in preventing 
the nuts from slipping out of their places, during the operation 
of describing the curve. 




Fig. 82. 



117. — To describe an ellipsis hy ordinates. Let a b and c a, 
{Fig. 82,) be given axes. With c e or e rf for radius, de- 
scribe the quadrant,/^ h ; divide f h, a e and e b, each into a 
like number of equal parts, as at ] , 2 and 3 ; through these 
points, draw ordinates, parallel to c d a.ndfg' ; take the distance, 
1 1, and place it at 1 /, transfer 2 ; to 2 m, and 3 ^ to 3 w ; through 
the points, a, w, m, / and c, trace a curve, and the ellipsis will 
be completed. 

The greater the number of divisions on a e, &c., in this and 
the following problem, the more points in the curve can be found, 
and the more accurate the curve can be traced. If pins are 
placed m the points, n^ m^ /, (fee, and a thin slip of wood bent 
around by them, the curve can be made quite correct. This 
method is mostly used in tracing face-moulds for stair hand- 
railing. 




118. — To describe an ellipsis by intersection of lines. V>o\ 



50 



PRACTICAL GEOMETRY. 



a h and c c?, [Fig. 83.) be given axes. Through t draw / ^, 
parallel tc a b ; from a and 6, draw a f and 6 ^, at right angles 
to ab ; divide f a, g b, a e and e b, each into a like number of 
equal parts, as at 1, 2, 3 and o, o, o ; from 1, 2 and 3, draw lines 
to c ; through o, o and o, draw lines from d, intersecting those 
drawn to c ; then a curve, traced through the points, «, i, i^ will 
^e that of an ellipsis. 




Fig. 84. 



Where neither trammel nor string is at hand, this, perhaps, Is 
the most ready method of drawing an ellipsis. The divisions 
should be small, where accuracy is desirable. By this method, 
an ellipsis may be traced without the axes, provided that a diame- 
ter and its conjugate be given. Thus, a b and c d, {Fig. 84,) are 
conjugate diameters : f g is drawn parallel to a 6, instead of 
being at right angles to c d ; also, / a and g b are drawn paraUel 
to c d, instead of being at right angles to a b. 




119.— To describe an ellipsis by intersecting arcs. I^et a b 



AMERICAN HOUSE-CARPENTER. 



5i 



mta c d, {Fig. 85,) be given axes. Between one of the foci,/ 
and/, and the centre, e, mark any number of points, at random, 
as 1, 2 and 3 ; upin/and/, with b 1 for radius, describe arcs at 
g, g, g and g ; upon/ and/, with a 1 for radius, describe arcs inter- 
secting the others at g^ g, g and g ; then these points of intersection 
will be in the curve of the elUpsis. The other points, h and i, are 
found in Uke manner, viz : h is found by taking b 2 for one radius, 
and a 2 for the other ; i is found by taking b 3 for one radius, and 
a 3 for the other, always using the foci for centres. Then by 
tracing a curve through the points, c, g, h, i, b, &c., the ellipse 
will be completed. 

This problem is founded upon the same principle as that of the 
string. This is obvious, when we reflect that the length of the 
string is equal to the transverse axis, added to the distance between 
the foci. See Fig. 80; in which c/ equals a e, the half of the 
transverse axis. 




120.— To descj^ibe a figure neai^ly in the shape of an ellip- 
sis^ by a pair of compasses. Let a b and c c?, {Fig. 86,) be 
given axes. From c, draw c e, parallel to ab ; from a, draw a e, 
parallel to c d; join e and d ; bisect e a in/; join / and c, inter- 
secting e d in i; bisect i cin o ; from o, draw og, at right angles 
to i c, meeting c d extended to g ; join i and g, cutting the trans- 
verse axis in r ; make h j equal to h g, and h k equal to h r , 
from J. through r and k, d'aw; m and/ n ; also, from g, through 
kf araw g I; upon g and ;, with g c for radius, describe the 



52 



PRACTICAL GEOMETRY. 



arcs, i I and m n ; upon r anl k^ with r a for radius, describe 
the arcs, 7n i and I n , this will complete the figure. 

When the axes are proportioned to one another as 2 to 3, the 
extremities, c and c?, of the shortest axis, will be the centres for 
describing the arcs, % /and ni n; and the intersection oi e d with 
the transverse axis, will be the centre for describing the arc, m i, 
&c. As the elliptic curve is coi;itinually changing its course from 
that of a circle, a true ellipsis cannot be described with a pair ot 
compasses. The above, therefore, is only an approximation. 




Fig. 87. 



121. — To draw an oval in the proportiofi, seven by nine. 
Let cd^ (^^^^- 87,) be the given conjugate axis. Bisect c g? in o, 
and through o, draw a 6, at right angles to c d ; bisect c o in e , 
upon 0, with o e for radius, describe the circle, e f g h ; from e, 
through h and/, draw e j and e i ; also, from g^ through h and/, 
draw g k and g I ; upon g^ with g c for radius, describe the arc, 
k I ; upon e, with e d for radius, describe the arc, j i ; upon h and 
/*, with k k for radius, describe the arcs, j k and I i; this will 
complete the figure. 

This is an approximation to an ellipsis ; and perhaps no 
method can be found, by which a well-shaped oval can be drawn 
with greater facility. By a little variation in the piocess, ovals 
of different proportions may be obtained. If quarter cf the trans- 
verse axis is taken for the radius of the circle, efg h, one will be 
drawn in the proportion, five by seven. 



AMERICAN HOUSE-CARPENTER. 



53 




122. — To draw a tangent to an ellipsis. Let abed, {Fig, 
88,) be the given ellipsis, and d the point of contact. Find the 
foci, {Art. 115,)/ and/, and from them, through d, draw/e and 
f d; bisect the angle, {Art. 77,) e d o, with the line, s r ; then 
s r will be the tangent required. 




Fig 89 



123. — An ellipsis with a tangent given, to detect the point 
of contact. Ijet agbf, {Fig. 89,) be the given ellipsis and tan- 
gent. Through the centre, e, draw a b, parallel to the tangent ; 
any where between e and/ draw c d, parallel to ab ; bisect cd in 
; through o and e, draw f g ; then g will be the point of con- 
tact required. 

124. — A diameter of an ellipsis given, to find its conjugate. 
Let a b, {Fig. 89,) be the given diameter. Find the line,/^, by 
the last problem ; then/^ will be th? diameter required. 



PRACTICAL GEOMETRY. 




125. — Any diameter and its conjugate being given, tc as- 
certain the two axes, and thence to describe the ellipsis, uet 
a b and c d, {Fig. 90,) be the given diameters, conjugate to one 
another. Through c, draw e /, parallel to a b ; from c, draw c 
g, at right angles to ef; make c g equal to a h or h b ; join g 
and h ; upon g, with g c for radius, describe the arc, i k c j ; 
upon hj with the same radius, describe the arc. In; through the 
intersections, I and ?^, draw n o, cutting the tangent, ef, in o ; 
upon 0, with o gfor radius, describe the semi-circle, eigf; join 
e and g, also g and /, cutting the arc, i c j, in k and t ; from e, 
through h, draw e m, also from/, through A, draw/p; from A; 
and ^, draw A; r and ^ 5, parallel to^ h, cutting e m in r, and/p 
in s ; make /t m equal to A r, and h p equal to h s ; then r m 
and 5 p will be the axes required, by which the ellipsis may be 
drawn in the usual way. 

126. — To describe an ellipsis, whose axes shall be propor- 
tiojiate to the axes of a larger o-^ smaller given one. Let a 
cbd, {Fig. 91,) be the given ellipsis and axes, and i j the trans- 
verse axis of a proposed smaller one. Join a and c ; trom i 
Iraw i e, parallel to a c ; make o / equal to o e ; then ef will be 



AMERICAN HOUSE-CARPENTER* 



55 




Fig. yi. 



the conjugate axis required, and will bear the same proportior. to 
j a.s c d does to a h. (See Art. 108.) 



12 3 Z 3 2 1 




1 2 3 m 3 
Fig. 92. 



127. — To describe a parabola by intersection of lines. Lee 
7n ?, {Fig. 92,) be the axis and height, (see Fig. 79,) and d d, a 
double ordinate and base of the proposed parabola. Through /. 
draw a a, parallel to d d ; through d and d, draw d a and d a, 
parallel to ni I ; divide a d and d ?n, each into a like number ol 
equal parts ; from each point of division in d m, draw the lines, 
1 1, 2 2, &c., parallel to ml; from each point of division in d 
a, draw lines to I ; then a curve traced through the points ol 
intersection, o, o and o, will be that of a parabola. 

127, a. — Another method. Let w. I, (Fig. 93,) be the axis and 
height, and d d the base. Extend m Z, and make I a equal to m 
t ; join a and o?, and a and d ; divide a d and a d. each into a 
liKe number of equal parts, as at 1, 2, 3, (fee ; join 1 and 1, 2 and 
2, (fee, and the parabola will be completed 



56 



PRACTICAL GEOMETRY. 



'/T ' 


^^1 


in/ 


\|j^S 


3/ 


^9 


a / 


\io 


/ 


\" 



Fig. 93. 




p I Z i 6 -Z 

Fig. 94. 



128. — To describe an hyperbola by intersection of lines. 
Let r 0, {Fig. 94,) be the height, p p the base, and n r the trans- 
verse axis. (See Fig. 79.) Through r, draw a a, parallel to jo 
f : irom p, draw a jo, parallel to r ; divide a p and p 0, each 
into a like number of equal parts ; from each of the points of di- 
visions in the base, draw lines to n ; from each of the points of 
division in a p, draw lines to r ; then a curve traced through the 
points of intersection, 0, 0, &c., will be that of an hyperbola. 

The parabola and hyperbola afford handsome curves for various 
mo ildings. 



de^io]N'stiiatio:ns. 



129. — To impress more deeply upon the mind of the learnei 
some of the more important of the preceding problems, and to 
indulge a very common and praiseworthy curiosity to discover 
the cause of things, are some of the reasons why the following 
exercises are introduced. In all reasoning, definitions are ne- 
cessary ; in order to insure, in the minds of the proponent and 
respondent, identity of idv?as. A corollary is an inference deduced 
from a previous course of reasoning. An as^iom it a proposition 
evident at first sight. In the following demonstrations, there are 
many axioms taken for granted ; (such as, things equal to the 
same thing are equal to one another, &c. ;) these it was thought 
not necessary to introduce in form. 




130. — Definition. If a straight line, as a b, {Fig. 95,) stand 

upon another straight line, as c d. so that the two angles made at 

8 



5S PRACTICAL GEOMETRY. 

the point. 6, are equal — a b c to a b d, (see note to Art. 27,) then 
each of the two angles is called a right angle. 

131. — Definition. The circumference of every circle is sup- 
posed to be divided into 360 equal parts, called degrees ; hence 
a semi-circle contains 180 degrees, a quadrant 90, &.c. 




132. — Definition. The measure of an angle is the number of 
degrees contained between its two sides, using the angular point 
as a centre upon which to describe the arc. Thus the arc, c e, 
{Fig. 96,) is the measure of the angle, c b e ; e a, of the angle, 
e b a ; and a d, of the angle, a b d. 

133. — Corollary. As the two angles at b, {Fig. 95,) are right 
angles, and as the serai-circle, cad, contains 180 degrees, {Art. 
131,) the measure of two right angles, therefore, is 180 degrees ; 
of oiije right angle, 90 degrees ; of half a right angle, 45 ; ot 
one-third of a right angle, 30, &c. 

134. — Definition. In measuring an angle, {Art. 132,) no re- 
gard is to be 11 ad to the length of its sides, but only to the degree 
of their inclination. Hence equal angles are such as have the 
same degree of inclination, without regard to the length of their 
sides. 




135. — Axiom. If two straight lines, parallel to one another, 



AMERICAN HOUSE-CARPENTER. 



59 



as a 6 and c d, {Fig. 97,) stand upon another straight line, as ef^ 
the angles, a bf and c d f, are equal : and the angle, a b e, is 
equal to the angle, c d e. 

136. — Definition. If a straight line, as a b, {Fig. 96,) stand 
obliquely upon anather straight line, as c o?, then one of the an- 
gles, as « 6 c, is called an obtuse a?igle, and the other, as a b d, 
an acute angle. 

137. — Axiom. The two angles, a b d and a b c, {Fig. 96,) are 
together equal to two right angles, {Art. 130, 133;) also, the 
three angles, a b d^e b a and c 6 e, are together equal to two right 
angles. 

138. — Corollary. Hence all the angles that can be made upon 
one side of a line, meeting in a point in that line, are together 
equal to two right angles. 

139. — Corollary. Hence all the angles that can be made on 
both sides of a line, at a point in that line, or all the angles thai 
can be made about a point, are together equal to four right angles. 




140. — Proposition. If to each of two equal angles a third 
angle be added, their sums will be equal.. Let a b cand d e f^ 
{Fig. 98,) be equal angles, and the angle, i j k, the one to be 
added. Make the angles, gb a and h e d, each equal to the given 
angle, ijk; then the angle, g b c. will be equal to the angle, h e 
f; for, \{ a b c and d e/be angles of 90 degrees, and i j k, 30; 
then the angles, g b c and h ef, will te each equal to 90 and 
30 added, viz : 120 degrees. 



60 



PRACTICAL GEOMETRY. 
i 




141.- -Proposition. Triangles that have two of their sides 
and the angle contained between them respectively equal, have 
also their third sides and the two remaining angles equal ; and 
consequently one triangle will every way equal the other. Let a 
b c, [Fig. 99,) and d efhe two given triangles, having the angle 
at a equal to the angle at d, the side, a b, equal to the side, d e, 
and the side, a c, equal to the side, df; then the third side of 
one, b c, is equal to the third side of the other, ef; the angle at b 
is equal to the angle at e, and the angle at c is equal to the angle 
at/. For, if one triangle be applied to the other, the three points, 
b, a, c, coinciding with the tliree points, e, d, /, the line, b c, must 
coincide with the line, e f; the angle at b with the angle at e ; % 
the angle at c with the angle at/; and the triangle, 6 a c, be every 
way eoual to the triangle, e df. 




142. — Proposition. The two angles at the base of an isoceles 
triangle are equal. Let a 6 c, {Fig. 100,) be an isoceles triangle, 
of which the sides, a b and a c, are equal. Bisect the angle, {Art, 



AMERICAN HOUSE-CARPENTER. 



C] 



77,) 6 a c, by the line, a d. Then the hne, b a, being equal tc 
the line, a c ; the line, a d, of the triangle, A, being equal to the 
line, a d, of the triangle, B, being conmion to each ; the angle, b 
a d^ being equal to the angle, d a c ; the line, b d, must, accord- 
ing to Art. 141, be equal to the hne, dc ; and the angle at b mus* 
be equal to the angle at c. 





Fiff. 101. 



143.— Proposition. A diagonal crossing a parallelogram di- 
vides it into two equal triangles. Let abed, (Fig. 101,) be a 
given parallelogram, and 6 c, a line crossing it diagonally. Then, 
as a c is equal to b d, and a b to c d, the angle at a to the angle 
at d, the triangle. A, must, according to Art. 141, be equal to the 
triangle, B. 




" /[. . ...IN 


A 


M 


/ ^ 


'V 


/'. / 



c i 'I <. id 

Fig. 102. 

144. — Proposition. Let abed, [Fig. 102,) be a given pa- 
rallelogram, and b ca diagonal. At any distance between a b and 
c d, draw e /, parallel to ab ; through the point, g, the intersection 
of the lines, b c and ef, draw h i, parallel to b d. In every paral- 
lelogram thus divided, the parallelogram, A, is equal to the paral- 
lelogram, B. According to Art. 143, the triangle, a b c, is 
equal to the triangle, bed; the triangle, Cj to the triangle, D ; 
and E to F ; this being the case, take D and P from the triangle, 
h c d, and C and E from the triangle, ab c^ and what remains 



PRACTICAL GF.OMETRY. 



in one must be ( q^ual to what remains in the other; therefore, tJie 
parallelogram, A, is equal to the parallelogram, B. 




Fig. 103. 



145. — Proj)osition. Parallelograms standing upon the same 
base and between the same parallels, are equal. Let ah c d and 
efcd^ {Fig. 103,) be given parallelograms, standing upon the 
same base, c d, and between the same parallels, a f and c d. 
Then, ab and e/ being equal to c d, are equal to one another; 
b e being added to both a b and e f^ a e equals b f; the line, ac, 
being equal to b d, and a e to b f, and the angle, c a e, being 
equal, {Art. 135,) to the angle, d b f, the triangle, a e c, must be 
equal, {Art. 141,) to the triangle, bfd; these two triangles being 
equal, take the same amount, the triangle, beg, from each, and 
what remains in one, a b g c, must be equal to what remains in .. 
the other, efdg; these two quadrangles being ^qual, add the 
same amount, the triangle, c g d, to each, and they must still be 
equal ; therefore, the parallelogram, ab c d, is equal to the paral- 
lelogram, efcd. 

146. — Corollary/. Hence, if a parallelogram and triangle stand 
upon the same base and between the same parallels, the parallelo- 
gram will be equal to double the triangle. Thus, the paral- 
lelogram, a d, {Fig. 103,) is double, {Art. 143,) the triangle, 
c e d. 

147. — Proposition. Let abed, {Fig. 104,) be a given quad- 
rangle with the diagonal, a d. From b, draw b e, parallel to a d, 
extend cdio e ; join a and e ; then the triangle, a e c, will be equal 
in area to the quadrangle, abed. Since the triangles, adh and 
a d e, stand uj on the same base, a d, and between the same parol 



AMERICAN HOUSE-CARPENTER. 63 




Irts, a d and b e, they are therefore equal, {Art. 145, 146 ;} and 
sinc^. th^ triangle, C, is common to both, the remaining triangles, A 
and B, are therefore equal ; then B being equal to A, the triangle, 
a e <;, is equal to the quadrangle, abed. 




148. — Proposition, If two straight lines cut each other, as 
a b and c d^ {Fig. 105,) the vertical, or opposite angles, A and 
C, are equal. Thus, a e, standing upon c c?, forms the angles, 
B and C, which together amount, {Art. 137,) to two right angles ; 
in the same manner, the angles, A and J5, form two right angles ; 
since the angles, A and B, are equal to B and C, take the same 
amount, the angle, B, from each pair, and what remains of one 
pair is equal to what remains of the other ; therefore, the an- 
gle, A, is equal to the angle, C. The same can be proved ot 
the opposite angles, B and D. 

149. — Proposition. The three angles of any triangle are 
equal to two right angles. Let a b c, {Fig. 106,) be a given tri- 
angle, with its sides extended to/, e, and d, and the line, eg, 



(^4 PRACTICAL GEOMETRY, 




drawn parallel to h e. As g c is parallel to e 6, the angle, g c d^ 
is, equal, {Art. 135,) to the angle, e h d ; as the lines, /c and b e, 
cut one another at a, the opposite angles, f a e and b a c, are 
equal, {Art. 148 ;) as the angle, /a e, is equal, (^r^. 135,) to the 
angle, a eg, the angle, a c ^, is equal to the angle, b a c ; there- 
fore, the three angles meeting at c, are equal to the three angles 
of the triangle, a b c ; and since the three angles ate are equal, 
{Art. 137,) to two right angles, the three angles of the triangle, a 
b c, must likewise be equal to two right angles. Any triangle 
can be subjected to the same proof. 

150. — Corollary. Hence, if one angle of a triangle be a right 
angle, the other two angles amount to just one right angle. 

151. — Corollary. If one angle of a triangle be a right angle, 
and the two remaining angles iire equal to one another, these are 
each equal to half a right angle. 

152. — Corollary. If any two angles of a triangle amount to 
a ri^ht angle, the remaining angle is a right angle. 

153. — Corollary. If any two angles of a triangle are togethei 
equal to the remaining angle, that remaining angle is a right 
angle. 

154. — Corollary. If any two angles of a triangle are each 
equal to two-thirds of a right angle, the remaining angle is alsc 
equal to two-thirds of a right angle. 

155. — Corollary. Hence, the angles of an equi-lateral trian 
gle. are each equal to two-thirds of a right angle. 



AMERICAN HOUSE-CiiRPENTER. 
6 




156. — Proposition. If from the extremities of the diameter of 
a semi-circle, two straight hnes be drawn to any point in tne cir- 
cumference, the angle formed by them at that point wiVi be a 
rignr angle. Let a 6 c, {Fig. 107,) be a given semi-circiC , and 
a arid b c, lines drawn from the extremities of the diameter, a 
c, to the given point, b ; the angle formed at that point by these 
lines, is a right angle. Join the point, b, and the centre, d ; the 
lines, d a, d b SLud d c, being radii of the same circle, are equal ; 
the angle at a is therefore equal, {Art. 142,) to the angle, ab d, 
also, the angle at c is, for the same reason, equal to the angie, d I 
c ; the angle, a 6 c, being equal to the angles at a and c taken 
together, must therefore, {Art. 153,) be a right angle. 




Fie. 108. 



J 57. — Proposition. The square of the hypothenuse of a 

right-angled triangle, is equal to the squares of the two remaining 

sides. Let a b c, {Fig. 108,) be a given right-angled triangle. 

having a square formed on each of its sides : then, the square, 6 e. is 

equal to the squares, A c and g 6, taken together. This can be 

9 



QQ PRACTICAL GEOMETRY. 

proved by showing that the parallelogram, b I, is equal to the squarei, 
q-b ; and that the parallelogram, c I, is equal to the square, h c. The 
angle, c 6 c?, is a right angle, and the angle, a 6/, is a right angle ; 
add to each of these the angle, ab c ; then the angle,/ b c, will evi- 
dently be equal, {Art. 140,) to the angle, ab d; the triangle,/^ c. 
and the square, g b, being both upon the same base, fb, and between 
the sitme parallels, fb and^ c, the square, g b. is equal, {Art. 146,) 
to twice the triangle./ 6 c ; the triangle, a b d, and the parallelo- 
gram, b I, being both upon the same base, b d, and between the 
same parallels, b d and a I, the parallelogram, b I, is equal to twice 
the triangle, a b d ; the triangles,/ 6 c and a b d, being equal to 
one another, {Art. 141,) the square,^ 5, is equal to the parallelo- 
gram, b I, either being equal to twice the triangle, /6 c or a b d. 
The method of proving h c equal to c / is exactly similar — thus 
proving the square, b e, equal to the squares, h c and g 6, taken 
together. 

This problem, which is the 47th of the First Book of Euclid 
is said to have been demonstrated first by Pythagoras. It is sta 
led, (but the story is of doubtful authority,) that as a thank-offer 
ing for its discovery he sacrificed a hundred oxen to the gods 
From this circumstance, it is sometimes called the liecato7nb pro- 
blem. It is of great value in the exact sciences, more especially 
in Mensuration and Astionomy, in which many otherwise intri- 
cate calculations are by it made easy of solution. 

158.. — Projjosition. In a segment of a circle, the versed sine 
equals the radius, less the square root of the difference of the 
squares of the radius and half-chord. That is, the versed sine, 
a c, {Fig. 109,) equals a 5, less c b. ISTow a h is radius, hence 
the radius, minus c 5, equals a c, the versed sine. To find the 
value of c b, it will be observed that c 5 is the side of the square, 
c f^ while the radius b d is the side of the square, b h, and the 
half-chord, c d, is the side of the square, c e ; also, that these 
three squares are made upon the three sides of the right angled 



AMEEICAiJ^ HOUSE-CAHPENTEE. 



tJ7 




Fig 109. 

triangle, h c d^ and the square, h A, is therefore equal to the two 
squares, c e and g /*, {Art. 157 ;) therefore, the square, c /*, is 
equal to the square, l A, minus the square, g e ; — or, is equal to 
the difference of the squares onh d and c d. Consequently the 
square root of g/ is equal to the square root of the difference 
of the squares on h d and g d; and since g h is the square root 
of c/", therefore g h equals the square root of the difference of 
the squares onh d and c d — or, equals the square root of the 
difference of the squares of the radius and the half- chord. 
Having found an expression for the value of g h, it remains 
merely to deduct this value from the radius, and the residue 
equals the versed sine ; for, as before stated, the versed sine, a c, 
equals the radius, a 5, minus Gh ; therefore, the versed sine 
equals the radius, minus the square root of the difference of 
the squares on the radius and half-chord. The rule expressed 
algebraically is v=r—Vr^—a''. where v is the versed sine, r the 
radius, and a the half-chord. It is read, v equals r, minus the 
square root of the difference of the squares of r and a. 

159. — Proposition. In an equilateral octagon the semi- 
diagonal of a circumscribed square, having its sides coincident 
with four of the s'des of the octagon, equals the distance al(>n^ 



68 



PRACTICAL GEOMETRY 




Fig. 110. 



a Side of the square from its corner to the more remote angle 
of the octagon occurring on that side of the square. To prove 
this, it need onlj to be shown that the triangle, a o d^ ^Fig. 
110,) is an isosceles triangle having its sides a o and a d, equal. 
The octagon being equi-lateral, it is also equi-angular, therefore 
the angles, h c o^ e c o^ a d o^ &c., are all equal. Of the right- 
angled triangle, y^ C')fc andfe being equal, the two angles, fe c 
andy*c e are equal, {Art. 142,) and are therefore, {Art, 151,) 
each equal to half a right angle. In like manner it may be 
shown that f al and/" h a are also each equal to half a right 
angle. And since fee and f ah are equal angles, therefore 
the lines e c and a h are parallel, {Art. 135,) and hence the 
angles, e c o and a o d^ are equal. These being equal, and the 
angles e c o and ado being, by construction, equal, as before 
shown, therefore the angles a o d and ado are equal, and con- 
sequently the lines a o and a d are equal. {Art. 142.) 

160. — Proposition. An angle at the circumference of a 
circle is measured by half the arc that subtends it ; that is, 
the angle ah c^ {Fig- HI?) is equal to half the angle a d c. 
Through the centre, d^ draw the diameter, h e. The triangle 
« 5 c? is an isosceles triangle, a d and h d being radii, and there- 
fore equal ; hence the two angles, d ah and dh a^ are equal, 



AMEKICAN HOUSE-CAEPENTEiC 




{Art. 142,) and the sum of these two angles is equal to the 
angle a d e, {Art. 149,) and therefore one of them, ah d, is 
equal to the half of a d e. The angles a d e and a h d {or 
a he) are both subtended by the arc a e. JSTow, since the angle, 
a d e, is measured by the arc a e, which subtends it, therefore 
the half of the angle, a d e, would be measured by the half 
of the arc a e; and since ah d is equal to the half of a d e, 
therefore ah d, or ah e, is measured by the half of the arc a e. 
It may be shown in like manner that the angle eh ois mea- 
sured by half the arc e c, and hence it follows that the angle, 
ah G^ is measured by half the arc, a c, that subtends it. 

161. — Proposition. In a circle, all the inscribed angles, 
ah G, {Fig. 112,) which stand upon the same side of the chord 
d e, are equal. For each angle is measured by half the arc 
df 6, {Art. 160,) hence the angles are all equal. 

162. — Corolla/ry. Equal chords, in the same circle, subtend 
equal angles. 

163. — Projposition. The angle formed by a chord and tan- 
gent is equal to any inscribed angle in the opposite segment 



TC 



PRACTICAL GEOMETEY. 




Fig. 112. 




Fig. 113. 



of the circle ; that is, the angle Z>, {Fig, 113,) equals the anglo 
A. Let of be the chord, and a h the tangent ; draw the dia- 
meter, d G j then <^ c & is a right angle, also d f c is a right 
angle. {Art. 156.) The angles A and B together equal a 
right angle, {Art, 150 ;) also the angles B and D together 
equal a right angle, (equal the angle d ch f) therefore the sum 
of A and B equals the sum of B and D. From each of these 
two equals, caking the like quantity B^ the remainders, J. and 



AMERICAN HOUSE-CARrENTEK. 



•ri 



Z), are equal. Thus, it is proved for tlie angle at d ; it is also 
true for any other angle ; for, since all other inscribed angles 
on that side of the chord line, c/", equal the angle A^ {Art. 
161,) therefore the angle formed by a chord and tangent equals 
any angle in the opposite segment of the circle. This being 
proved for the acute angle, 7>, it is also true for the obtuse 




angle, a c f j for, from any point, n^ (Fig. 114-,) in the arc 
c nf^ draw lines to <^,y and c ; now, if it can be proved t''^«t 
the angle a cf equals the angle f n c, the entire proposition 
is proved, for the angle f n c equals any of all the inscribed 
angles that can be drawn on that side of the chord. {Art. 
161.) To prove, then, that a of equals c nf: the angle a cf 
equals the sum of the angles A and B / also the angle c nf 
equals the sum of the angles C and D. The angles B and Z>, 
being inscribed angles on the same chord, df^ are equal. The 
angles G and A being right angles, [Art. 156,) are likewise 
equal. E'ow, since A equals G^ and B equals D., therefore 
the sum of A and B equals the sum of G and D — or the angle 
a of equals the angle g nf. 

164. — Proposition. Two chords, a h and c d, {Fig. 115,) 
intersecting, the parallelogram or rectangle formed by the two 
parts of one is equal to the rectangle formed by the two parts 
of the other. That is, c e mi Itiplied by e d, the product ia 



72 



PKACTICAL GEOMETRY. 




Fig. 115. 



equal to the product of a e multiplied by eh. The triangle A 
is similar to the triangle B^ because it has corresponding an- 
gles. The angle i equals the angle e^ [Art 148 ;) the angle at 
c equals the angle at a because they stand upon the same 
chord, d 5, {Art 161 ;) for the same reason the angle h equals 
the angle d^ for ^ach stands upon the same chord, a c. There- 
fore, the triangle A having the same angles as the triangle B^ 
the length of the sides of one are in like proportion as the 
length of the sides in the other. So, e d : a e y, e h I o e. 
Hence, a e multiplied by 6 Z» is equal to e d mutiplied by c e — 
or the product of the means equals the product of the ex 
tremes. 

165.-- -Proposition. In any circle, when a segment is given, 
the radius is equal to the sum of the squares of half the chord 
and of the versed sine, divided by twice the versed sine. Let 
a J, [Fig. 116,) be the chord line, and v the versed sine of the 
segment. By the preceding article the triangle A is shown to 
be like the triangle B^ having equal angles and proportionate 



AMERICAN HOUSE-CARPENTEK. 



73 




n' 



IcEcth of sides. Therefore, -y : ^::m : ^', or — = i : that is, i ia 

V 

equal to the square of n (or n x n) divided by v. This result 
being added to v equals the diameter o x, which may be indi 

cated by the letter d ; thus, — -\- v ~i-\- v — d ; and the half 



n' 



of this, or- 



d 



— — = — = r — the radius. Reducing this expres- 

2i 2 



sion by multiplying the numerator and denominator each by the 

^2 _|_ ^2 

like quantity, viz. -y, there results, — = r; and where c 

represents the chord, the expression is, ^^^ = r : that is, 

as stated above, the radius is equal to the sum of the squares 
of half the chord and of the versed sine, divided by twice the 
versed sine. 

166. — Proposition. Any ordinate, m n^ {Fig. 117,) in thb 
eegment of a circle, is equal to the square root of the difference 

10 



PEACTICAL GEOMETEY. 




of the squares of the radius and abscissa, {d n^) less the di£fer- 
ence of the radius and versed sine. So, if the chord a J, and 
the versed sine c d^ be given, the length of any number of 
ordinates may be found by which to describe the arc. Find 
the radius, c e^ by the preceding Article. It will be observed 
that e m is also radius. Then, to find the length of the ordi- 
nate, m n, make e o equal to d n : now, according to Article 
157, the square of e o taken from the square of e m^ the residue 
equals the square of o m, and the square root of this residue 
will be the length of the line o m. Tlien from o m take o n 
equal to e d^ and the result will be the length of m n. That is, 
the ordinate is equal to the square root of the difference of the 
squares of the radius and abscissa, less the difference of the 
radius and versed sine. This may be expressed algebraically 
thus : y = -^r' — x' — {r — v), where y is the ordinate, r the 
radius, x the abscissa, and v the versed sine ; — d n being the 
abscissa of the ordinate n m^ d g the abscissa of the ordinate 



AMERICAN HOUSE-OARPENTEK. Y5 

9f') <^c. : tlie abscissa being in each case the distance from the 
foot of the versed sine, g d^ to the foot of the ordinate whose 
length is sought. 




16Y. — Projjosition. The sides of any quadrangle being 
bisected, and lines drawn joining the points of bisection in the 
adjacent sides, these lines will form a parallelogram. Draw 
the diagonals, Oj h and c d^ {Fig. 118.) It will here be per- 
ceived that the two triangles, a e o and a c d^ are homologous, 
having like angles and proportionate sides. Two of the sides 
of one triangle lie coincident with the two corresponding sides 
of the other triangle, therefore the contained angles between 
these sides in each triangle are identical. By construction, 
these corresponding sides are proportionate ; a c being equal 
to twice a e, and a d being equal to twice a o ; therefore the 
remaining sides are proportionate, c d being equal to twice e 6>, 
hence the remaining corresponding angles are equal. Since, 
then, the angles a e o and a g d are equal, therefore the line e o 
is parallel with the diagonal g d — so, likewise, the line m nh 
parallel to the same diagonal, g d. If, therefore, these two 
lines, e o and m ^, are parallel to the same line, c d, they must 
be parallel to each other. In the same manner the lines o n 
and e m are proved parallel to the dii^gonal, a h, and to each 



^^ PEACTICAL GEOMETET. 

other ; therefore the inscribed figure, m e o n, is a paralleic- 
gram. It may be remarked also, that the parallelogram so 
formed will contaii just one-half the area of the circumscribing 
quadrangle. 



These demonstrations, which relate mostly to the problems 
previously given, are introduced to satisfy the learner in regard 
to their mathematical accuracy. By studying and thoroughly 
understanding them, he will soonest arrive at a knowledge of 
their im^^ortance, and be likely the longer to retain them in 
memory. Should he have a relish for such exercises, and wish 
to continue them farther, he may consult Euclid's Elements, in 
which the whole subject of theoretical geometry is treated of 
in a manner sufficiently intelligible to be understood by the 
young mechanic. The house-carpenter, especially, needs infor- 
mation of this kind, and were he thoroughly acquainted with 
the principles of geometry, he would be much less liable to 
commit mistakes, and be better qualified to excel in the execu- 
tion of iis often difficult undertakings. 



8^XT1().\ Ji.— ARCHTTECTTTRK. 



HISTORY OF ARCHITECTURE. 

168. — Architecture has been defined to be — "the art of build 
ing ;" but, in its common acceptation, it is — " the art of designing 
and constructing buildings, in accordance with such principles as 
constitute stability, utility and beauty." The literal signification 
of the Greek word archi-tecton^ from which the word architect 
is derived, is chief-carpenter ; but the architect has always been 
known as the chief designer rather than the chief builder. Of 
the three classes into which architecture has been divided — viz., 
Civil, MiUtary, and Naval, the first is that which refers to che 
construction of edifices known as dwellings, churches and other 
public buildings, bridges, &c., for the accommodation of civilized 
man — and is the subject of the remarks which follow. 

169. — This is one of the most ancient of the arts: the scrip- 
tures inform us of its existence at a very early period. Cain, 
the son of Adam, — " builded a city, and called the name of the 
city after the name of his son, Enoch" — but of the peculiar style 
or manner of building we are not informed. It is presumed that 
it was not remarkable for beauty, but that utility and perhaps sta- 
bility were its characteristics. Soon after the deluge — that me 



78 AMERICAN HOUSF-CARPENTER. 

morable event, which removed from existence all traces of the 
works of man — the Tower of Bahel was commenced. This was 
a work of such magnitude that the gathering of the materials, 
according to some writers, occupied three years ; the period from 
its commencement until the work was abandoned, was twenty- 
two years ; and the bricks were like blocks of stone, being twenty 
feet long, fifteen broad and seven thick. Learned men have ^ven 
it as their opinion, that the tower in the temple of Behis at Baby 
Ion was the same as that which in the scriptures is called the 
Tower of Bab^l. The tower of the temnle of Belus was square 
at its base, eacn siae measurmg one lurlong, and consequently 
half a mile in circumference. Its form was that of a pyramid 
and its height was 660 feet. It had a winding passage on the 
outside from the base to the summit, which was wide enough for 
two carriages. 

ITO.— Historical accounts of ancient cities, of which there are 
.jow but few remains — such as Babylon, Palmyra and Ninevah 
of the Assyrians ; Sidon, Tyre, Aradus and Serepta of the Phoe- 
nicians ; and Jerusalem, with its splendid temple, of the Israelites 
— show that architecture among them had made great advances. 
Ancient monuments of the art are found also among other nations ' 
the subterraneous temples of the Hindoos upon the islands, Ele- 
phanta and Salsetta ; the ruins of Persepolis in Persia ; pyramids, 
obelisks, temples, palaces and sepulchres in Egypt — all prove that 
the architects of those early times were possessed of skill and 
judgment highly cultivated. The principal characteristics of 
their works, are gigantic dimensions, immoveable solidity, and, in 
some instances, harmonious splendour. The extraordinary size 
of some is illustrated in the pyramids of Egypt. The largest of 
these stands not far from the city of Cairo : its base, which is 
square, covers about 11| acres, and its height is nearly 500 feet 
The stones of which it is built are immense — the smallest being 
full thirty feet long. 

171 .--Among the Greeks, architecture was cultivated as a fine 



ARCHITECTURE. 



79 



art, and rapidly advanced towards perfection. Dignity and grace 
were added to stability and magnificence. In the Doric order, 
their first style of building, this is fully exemplified. Phidias. 
Tctinus and Oallicrates, are spoken of as masters in the art at this 
period: the encouragement and support of Pericles stimulated 
them to a noble emulation. The beautiful temple of Minerva, 
erected upon the acropolis of Athens, the Propyleum, the Odeum 
and others, were lasting monuments of their success. The Ionic 
and Corinthian orders were added to the Doric, and many mag- 
nificent edifices arose. These exemplified, in their chaste propor- 
tions, the elegant refinement of Grecian taste. Improvement in 
Grecian architecture continued to advance, until perfection seems 
to have been attained. The specimens which have been partially 
preserved, exhibit a combination of elegant proportion, dignified 
simplicity and majestic grandeur. Architecture among the 
Greeks was at the height of its glory at the period immediately 
preceding the Peloponnesian war ; after which the art declined. 
An excess of enrichment succeeded its former simple grandeur ; 
yet a strict regularity was maintained amid the profusion of orna- 
ment. After the death of Alexander, 323 B. C, a love of gaudy 
splendour increased : the consequent decline of the art was 
visible, and the Greeks afterwards paid but little attention to the 
science. 

172. — While the Greeks were masters in architecture, whicti 
they applied mostly to their temples and other public buildings, 
the Romans gave their attention to the science in the construction 
of the many aqueducts and sewers with which Rome abounded : 
building no such splendid edifices as adorned Athens, Corinth 
and Ephesus. until about 2)0 years B. C, when their intercourse 
with the Greeks became more extended. Grecian architecture 
was introduced into Rome by Sylla ; by whom, as also by Marius 
and Caesar, many large edifices were erected in various cities of 
Italy. But under Caesar Augustus, at about the beginning of the 
christian era, the art arose to the greatest perfection it ever at- 



80 AMERICAN HOUSE-CARPENTER. 

tained in Italy. Under his patronage. Grecian artists were en- 
couraged, and many emigrated to Rome. It was at about thw 
time that Solomon's temple at Jerusalem was rebuilt by Herod — 
a Roman. This was 46 years in the erection, and was most pro 
bably of the Grecian style of building — perhaps of the Corin- 
thian order. Some of the stones of which it was built were 46 
feet long, 21 feet high and 14 thick ; and others were of the 
astonishing length of 82 feet. The porch rose to a great height ; 
the whole being built of white marble exquisitely polished. This 
is the building concerning which it was remarked — " Master, see 
what manner of stones, and what buildings are here." For the 
construction of private habitations also, finished artists were em- 
ployed by the Romans : their dwellings being often built with the 
finest marble, and their villas splendidly adorned. After Augus- 
tus, his successors continued to beautify the city, until the reign of 
Constantine ; who, having removed the imperial residence to 
Constantinople, neglected to add to the splendour of Rome ; and 
the art, in consequence, soon fell from its high excellence. 

Thus we find that Rome was indebted to Greece for what she 
possessed of architecture — not only for the knowledge of its prin- 
ciples, but also for many of the best buildings themselves ; these 
having been originally erected in Greece, and stolen by the un- 
prnicipled conquerors — taken down and removed to Rome. 
Gieece was thus robbed of her best monuments of architecture. 
Touched by the Romans, Grecian architecture lost much of its 
elegance and dignity. The Romans, though justly celebrated 
for their scientific knowledge as displayed in the construction of 
their various edifices, were not capable of appreciating the simple 
grandeur, the refined elegance of the Grecian style ; but sought 
to improve upon it by the addition of luxurious enrichment, and 
thus deprived it of true elegance. In the days of Nero, whose 
palace of gold is so celebrated, buildings were lavishly adorned. 
Adrian did much to encourage the art ; but not satisfied with the 
simplicity of the Grecian style, the artists of his time aimed at 



ARCHITECTURE. 81 

inventing nevf ones, and added to the already redundant embel- 
lishments of the previous age. Hence the origin of the pedestal, 
the great variety of intricate ornaments, the convex frieze, the 
round and the open pediments, &c. The rage for luxury 
continued until Alexander Severus, who made some impDve- 
ment; but very soon after his reign, the art began rapidly to 
decline, as particularly evidenced in the mean and trifling charac- 
ter of the ornaments. 

173. — The Goths and Yandals, when they overran the coun- 
tries of Italy, Greece, Asia and Africa, destroyed most of the 
works of ancient architecture. Cultivating no art but that of 
war, these savage hordes could not be expected to take any interest 
in the beautiful forms and proportions of their habitations. From 
this time, architecture assumed an entirely different aspect. The 
celebrated styles of Greece were unappreciated and forgotten ; and 
modern architecture took its first step on the platform of existence. 
The Goths, in their conquering invasions, gradually extended it 
over Italy, France, Spain, Portugal and Germany, into England. 
From the reign of Gallienus may be reckoned the total extinction 
of the arts among the Romans. From his time until the 6th or 
7th century, architecture was almost entirely neglected. The 
buildings which were erected during this suspension of the arts. 
were very rude. Being constructed of the fragments of the edi- 
fices which had been demolished by the Visigoths in their unre- 
strained fury, and the builders being destitute of a proper know- 
ledge of architecture, many sad blunders and extensive patch- 
work might have been seen in their construction — entablatures 
inverted, columns standing on their wrong ends, and other ridi- 
culous arrangements characterized their clumsy work. The vast 
number of columns which the ruins around them afforded, they 
used as piers in the construction of arcades — which by some is 
thought, after having passed through various changes, to have 
been the origin of Ihe plan of the Gothic cathedral. Buildhigs 
generally, which ar3 not of the classical styles, and which were 

11 



82 AMERICAN HOUSE-CARPENTER. 

erected after the fall of the Roman empire, have by some been 
iiidiscriminately included under the term Gothic. But the 
changes which architecture underwent during the dark ages, show 
that there were several distinct modes of building. 

174. — Theodoric, king of the Ostrogoths, a friend of the arts, 
who reigned in Italy from A. D. 493 to 525, endeavoured to re- 
store and preserve some of the ancient buildings ; and erected 
others, the ruins of which are still seen at Verona and Ravenna. 
Simplicity and strength are the characteristics of the structures 
erected by him ; they are, however, devoid of grandeur and ele- 
gance, or fine proportions. These are properly of the Gothic 
style ; by some called the old Gothic to distinguish it from the 
pointed style, which is generally called modern Gothic. 

175. — The Lombards, who ruled in Italy from A. D. 568, had 
no taste for architecture nor respect for antiquities. Accordingly, 
they pulled down the splendid monuments of classic architecture 
which they found standing, and erected in their stead huge build- 
ings of stone which were greatly destitute of proportion, elegance 
or utility — their characteristics being scarcely any thing more than 
stability and immensity combined with ornaments of a puerile cha- 
racter. Their churches were disfigured with rows of small columns 
along the cornice of the pediment, small doors and windows with 
circular heads, roofs supported by arches having arched buttresses 
to resist their thrust, and a lavish display of incongruous orna- 
ments. This kind of architecture is called, the Lombard style, 
and was employed in the 7th century in Pa via, the chief city of 
the Lombards ; at which city, as also at many other places, a 
great many edifices were erected in accordance with its inelegant 
forms. 

176. — The Byzantine architects, from Byzantium, Constantino- 
ple, erected many spacious edifices ; among which are included 
the cathedrals of Bamberg, Worms and Mentz, and the most an 
cient part of the minster at Strasburg ; in all of these they com- 
bined the Roman-Ionic order with the Gothic of the Lombards. 



ARCHITECTURE. od 

This Style is called the Lombard-Byzantine. To the last style 
there were afterwards added cupolas similar to thost used in the 
east, together with numerous slender pillars with tasteless capi- 
tals, and the many minarets which are the characteristics of the 
proper Byzantine^ or Oriental style. 

177. — In the eighth century, when the Arabs and Moors de- 
stroyed the kingdom of the Goths, the arts and sciences were 
mostly in possession of the Musselmen-conquerors ; at which 
time there were three kinds of architecture practised ; viz : the 
Arabian, the Moorish and the modern-Gothic. The Arabian 
style was formed from Greek models, having circular arches 
added, and towers which terminated with globes and minarets. 
The Moorish is very similar to the Arabian, being distinguished 
from it by arches in the form of a horse-shoe. It originated in 
Spain in the erection of buildings with the ruins of Roman archi- 
tecture, and is seen in all its splendour in the ancient palace of the 
Mohammedan monarchs at Grenada, called the Alhambra^ or red- 
house. The Modern-Gothic was originated by the Visigoths 
in Spain by a combination of the Arabian and Moorish styles ; 
and introduced by Charlemagne into Germany. On account of 
the changes and improvements it there underwent, it was, at about 
the 13th or 14th century, termed the German^ or romantic style. 
It is exhibited in great perfection in the towers of the minster of 
Strasburgh, the cathedral of Cologne and other edifices. The 
most remarkable features of this lofty and aspiring style, are the 
lancet or pointed arch, clustered pillars, lofty towers and flying 
buttresses. It was principally employed in ecclesiastical archi- 
tecture, and in this capacity introduced into France, Italy, Spain, 
and England. 

178. — The Gothic architecture of England is divided into the 
Norman^ the Early -Eiiglish^ the Decorated^ and the Perpen- 
dicular styles. The Norman is principally distinguished by the 
character of its ornaments — the chevron, or zigzags being the 
most common. Buildings in this style were erected in the 12tb 



84 



AMERICAN HOUSE-CARPENTER. 



century. The Early-English is celebrated for the beauty of its 
edifices, the chaste simplicity and purity of design which they 
display, and the peculiarly graceful character of its foliage. This 
style is of the IHth century. The Decorated style, as its name 
implies, is characterized by a great profusion of enrichment, 
which consists principally of the crocket, or feathered-ornament, 
and ball -flower. It was mostly in use in the 14th century. The 
Perpendicular style, which dates from the 15th century, is distin- 
guished by its high towers, and parapets surmounted with spires 
similar in number and grouping to oriental minarets. 

179. — Thus these several styles, which have been erroneously 
termed Gothic^ were distinguished by peculiar characteristics as well 
as by different names. The first symptoms of a desire to return to a 
pure style in architecture, after the ruin caused by the Goths, was 
manifested in the character of the art as displayed in the church 
of St. Sophia at Constantinople, which was erected by Justinian 
in the 6th century. The church of St. Mark at Venice, which 
arose in the 10th or 11th century, was the work of Grecian archi- 
tects, and resembles in magnificence the forms of ancient archi- 
tecture. The cathedral at Pisa, a wonderful structure for the age, 
was erected by a Grecian architect in 1016. The marble with 
which the walls of this building were faced, and of which the four 
rows of columns that support the roof are composed, is said to be 
of an excellent character. The Campanile, or leaning-tower as it 
is usually called, was erected near the cathedral in the 1.2th cen- 
tury. Its inclination is generally supposed to have arisen from 
a poor foundation ; although by some it is said to have been thus 
constructed originally, in order to inspire in the minds of the 
beholder sensations of sublimity and awe. In the 13th century, 
the science in Italy was slowly progressing ; many fine churches 
were erected, the style of which displayed a decided advance m 
the progress towards pure classical architecture. In other parts 
of Europe, the Gothic, or pointed style, was prevalent. The 
cath 'dral at Strasburg, designed ' y Irwin Steinbeck, was erected 



AUCHITECTURE. . 85 

in tlie 13th and 14:th centuries. In France and England dur- 
ing tlie 14th century, many very superior edifices were erected 
in this style. 

180. — In the 14:th and 15th centuries, and particularly in the 
latter, architecture in Italy was greatly revived. The masters 
began to study the remains of ancient Roman edifices ; and many 
splendid buildings were erected, which displayed a purer taste 
in the science. Among others, St. Peter's of Rome, which was 
built about this time, is a lasting monument of the architectural 
skill of the age. Giocondo, Michael Angelo, Palladio, Yignola, 
and other celebrated architects, each in their turn, did much to 
restore the art to its former excellence. In the edifices which 
were erected under their direction, however, it is plainly to be 
seen that they studied not from the pure models of Greece, but 
from the remains of the deteriorated architecture of Rome. The 
high pedestal, the coupled columns, the rounded pediment, the 
many curved-and-twisted enrichments, and the convex frieze, 
were unknown to pure Grecian architecture. Yet their efifbrte 
were serviceable in correcting, to a good degree, the very 
impure taste that had prevailed since the overthrow of the Ro- 
man empire. 

181. — At about this time, the Italian masters and numerous 
artists who had visited Italy for the purpose, spread the Roman 
style over various countries of Europe ; which was gradually re- 
ceived into favor in place of the modern-Gothic. This fell into 
disuse ; although it has of late years been again cultivated. It 
requires a building of great magnitude and complexity for a per- 
fect display of its beauties. In America, the pure Grecian style 
was at first more or less studied ; and perhaps the simplicity of 
its principles would be better adapted to a republican country, 
than the intricacy and extent of those of the Gothic ; but at the 
present time the latter style is being introduced, especially foi 
eccJesiastical structures. 



86 AMERICAN HOrSE-CARPENTER. 

STYLES OF ARCHITECTrRE. 

182. — It is generally acknowledged that the yarions styles in 
architecture, were originated in accordance with the diflenmt 
pursuits of the early inhabitants of the earth ; and were brought 
by their descendants to their present state of perfection, throiiglF 
the propensity for imitation and desire of emnlation which are 
found more or less among all nations. Those that followed 
agricultural pursuits, from being employed constantly upon 
the same piece of land, needed a permanent residence, and the 
wooden hut was the offspring of their wants ; while the shep- 
herd, who followed his fiocks and was compelled to trayerse 
large tracts of country for pasture, found the tent to be the 
most portable habitation ; again, the man deyoted to hunting 
and fishing — an idle and yagabond way of Hying — is naturally 
supposed to haye been content with the cavern as a place of 
shelter. The latter is said to haye been the origin of the 
Egyptian style ; while the curyed roof of Chinese structures 
giyes a strong indication of their haying had the tent for their 
model ; and the simplicity of the original style of the Greeks, 
(the Doric,) shows quite conclusiyely, as is generally conceded, 
that its original was of wood. The modern-Gothic, or pointed 
style, which was most generally confined to ecclesiastica* 
structures, is said by some to haye originated in an attempt to 
imitate the bower, or grove of trees, in which the ancients per- 
formed their idol-worship. 

183. — There are numerous styles, or orders, in architecture , 
and a knowledge of the peculiarities of each is important to the 
student in the art. An Order, in architecture, is composed of 
three principal parts, yiz : the Stylobate, the Column and the 
Entablature. 

184. — ^The Stylobate is the substructure, or basement, upon 
which the columns of an order are arranged. In Roman archi- 
tecture — especially in the interior of an edifice — it frequently 
occurs that each column has a separate substruct'ire ; this ia 



AECHITECTUEE. 87 

called Si pedestal. If possible, the pedestal should be avoided 
in all cases ; because it gives to the column, the appearance of 
having been originally designed for a small building, and after* 
wards pieced-out to make it long enough for a larger one. 

185. — The Column is composed of the base, shaft and capital. 

186. — The Entablatui'.e, above and supported by the co- 
lumns, is horizontal ; and is composed of the architrave, frieze 
and cornice. These principal parts are again divided into 
various members and mouldings. (See Sect. III.) 

187. — The Base of a column is so called from hasis^ a found- 
ation, or footing. 

188. — The Shaft, the upright part of a column standing 
upon the base and crowned with the capital, is from shafto^ to 
dig — in the manner of a well, whose inside is not unlike the 
form of a column. 

189. — The Capital, from hephale or caput^ the head, is the 
uppermost and crowning part of the column. 

190. — The Architrave, from arehi^ chief or principal, and 
trahs^ a beam, is that part of the entablature which lies in 
immediate connection with the column. 

191. — The Frieze, from fibron^ a fringe or border, is that 
part of the entablature which is immediately above the archi- 
trave and beneath the cornice. It was called by some of the 
ancients, zophorus^ because it was usually enriched with sculp- 
tured animals. 

192. — The Cornice, from corona^ a crown, is the upper and 
projecting part of the entablature — being also the uppermost 
and crowning part of the whole order. 

193. — The Pediment, above the entablature, is the triangular 
portion which is formed by the inclined edges of the roof at 
the end of the building. In Gothic architecture, the pediment 
is called, a gable, 

194. — ^Tlie Tympanum is the perpendicular triangular surface 
which is enclosed by the c<i mice of the pediment. 



88 AMEBIC AN^ HOUSE-CABPENTEJR. 

195. — The Attic is a small order, consisting of pilastei*s and 
entablature, raised above a l-arger order, instead of a pediment 
An attic storj is the iippei- story, its windows being usually 
square. 

196. — An order, in architectm-e, has its several parts and 
members proportioned to one another by a scale of 60 equal 
parts, which are called' minutes. If the height of buildings 
were always the same, the scale of equal parts would be a 
hxed quantity — an exact number of feet and inches. But as 
buildings are erected of different heights, the column and its 
accompaniments are required to be of different dimensions. 
To ascertain the scale of equal parts, it is necessary to know 
the height to which the whole order is to be erected. This 
must be divided by the number of diameters which is directed 
for the order under consideration. Then the quotient obtained 
by such division, is the length of the scale of equal parts — and 
is, also, the diameter of the column next above the base. For 
instance, in the Grecian Doric order the whole height, includ- 
ing column and entablature, is 8 diameters. Suppose now it 
were desirable to construct an example of this order, forty feet 
high. Then 40 feet divided by 8, gives 5 feet for the length 
of the scale ; and this being divided by 60, the scale is com- 
pleted. The upright columns of figures, marked H and P^ by 
the side of the drawings illustrating the orders, designate the 
height and the projection of the members. The projection of 
each member is reckoned from a line passing through the axis 
of the column, and extending above it to the top of the enta- 
blature. The figures represent minutes, or 60ths, of the major 
diameter of the shaft of the column. 

197. — Gkecian Styles. The original method of building 
among the Greeks, was in what is called the Doric order : to 
this were afterwards added the lonio and the Corinthian, 
These three were the only styles known among them. Each is 
distingiishfd from the other two, by not only a peculiarity of 



AECHITECTUEE. O^ 

some one or more of its principal parts, but also hy a particular 
destination. The character of the Doric is robust, manly and 
Eerculean-like ; that of the Ionic is more delicate, feminine, 
matronly ; wliile that of the Corinthian is extremely delicate, 
)routhful and virgin-like. However they may differ in theii 
general character, they are alike famous for grace and dignity, 
elegance and grandeur, to a high degree of perfection. 

198. — The DoKic Oedek, {J^ig. 120,) is so ancient that its origin 
is unknown — although some have pretended to have discovered 
it. But the most general opinion is, that it is an improvement 
upon the original wooden buildings of the Grecians. These no 
doubt were very rude, and perhaps not unlike the followin^^ 
figure. 




Fig. 119. 

The trunks of trees, set perpendicularly to support the roof, 
may be taken for columns ; the tree laid upon the tops of the 
perpendicular ones, the architrave ; the ends of the cross-beams 
which rest upon the architrave, the triglyphs ; the tree laid on 
the cross-beams as a support for the ends of the rafters, the 
bed-moulding of the cornice ; the ends of the rafters which 
project beyond the bed-moulding, the mutules; and perhaps 
the projection of the roof in front, to screen the entrance from 
the weather, gave origin to the portico. 

The peculiarities of the Doric order are the triglyphs — those 

parts of the frieze which have perpendicular channels cut in 

12 



DORIC ORDER. 




Fig 120 



ARCHITECTURE. 91 

tneir surface ; tlie absence of a base to the column — as alco of 
fillets between the iiutings of the column, and the plainness of 
the capital. The triglyphs are to be so disposed that the width 
of the metopes — the spaces between the triglyphs — shall be 
equal to their height. 

199. — ^The intercolumniation^ or space between the columns, 
is regulated by placing the centres of the columns under the 
centres of the triglyphs — except at the angle of the building ; 
where, as may be seen in Fig. 120, one edge of the triglyph 
must be over the centre of the column.* "Where the columns 
are so disposed that one of them stands beneath every other tri- 
glyph, the arrangement is called, mono-triglyph^ and is most 
common. When a column is placed beneath every third tri- 
glyph, the arrangement is called diastyle / and when beneath 
every fourth, arcwstyle. This last style is the worst, and is sel- 
dom adopted. 

200. — The Doric order is suitable for buildings that are des- 
tined for national purposes, for banking-houses, &c. Its ap- 
pearance, though massive and grand, is nevertheless rich and 
graceful. The Patent Office at Washington, and the Custom - 
House at New York, are good specimens of this order. 



^ Grecian Doric Order. When the width to be occupied by the whole front is limited; to deter' 
mine the diameter of the column. 
The relation between the parts may be expressed thus : 

^a 

^~d{b + c)-[-{m — c) 
Where a equals the width in feet occupied by the columns, and their intercolumniations takec 
coliectively, measus-ed at the base ; b equals the width of the metope, in minutes ; c equals the width 
of the triglyphs in minutes; d equals the number of metopes, and x equals the diameter in feet. 

Example. — A front of six columns— hexastyle — 61 feei wide ; the frieze having one triglyph over 
each intercolumniation, or mono-triglyph. In this case, there being five intercolumniations and two 
metopes over each, therefore there are 5 X 2= 10 metopes. Let the metope equal 42 minutes and 
the triglyph equal 38. Then a = G 1 ; 6 =r 42 ; c = 28 ; and rf = 10 ; and the formula above becomes, 
60X61 60X61 3660 

' = m¥z + W+W=^) = i^3r70+^= 732"= ^ ^^^' =" '^^ ^''''^^'^' "- i^"^- 
Exampte.—An octastyle front, 8 columns, 184 feet wide, three metopes over each intercolurania 
ttoii, 21 in all, and the metope and triglyph 42 anc 28, as before. Then. 

60X184 11040 ,,o- 30 . . *u ^• 

" = 2U42T28H-l60^28T = l502-= '^""^^h ^^^^ = the diameter required. 



lOmO OBBEK 




Fi«. 121. 



AECHITECTUEE. 93 

201.— Tlie Ionic Order. {Fig. 121.) The Doric was for 
Bome time the only order in use among the Greeks. They gave 
their attention to the cultivation of it, until perfection seems to 
have been attained. Their temples were the principal objects 
upon which their skill in the art was displayed; and as the 
Doric order seems to have been well fitted, by its massive pro- 
portions, to represent the character of their male deities rather 
than the female, there seems to have been a necessity for an- 
other style which should be emblematical of feminine graces, 
and with which they might decorate such temples as were de- 
dicated to the goddesses. Hence the origin of the Ionic order. 
This was invented, according to historians, by Hermogenes of 
Alabanda ; and he being a native of Caria, then in the posses- 
sion of the lonians, the order was called, the Ionic. 

202. — ^The distinguishing features of this order are the vo 
lutes^ or spirals of the capital ; and the dentils among the bed- 
mouldings of the cornice : although in some instances, dentils 
are wanting. The volutes are said to have been designed as a 
representation of curls of hair on the head of a matron, of whom 
the whole column is taken as a semblance. 

203. — The intercolumniation of this and the other orders — 
both Eoman and Grecian, with the exception of the Doric- 
are distinguished as follows. When the interval is one and a 
half diameters, it is called, pycnostyle^ or columns thick-set ; 
when two diameters, systyle / when two and a quarter diame- 
ters, eicstyle ; when three diameters, diastyle ; and when more 
than three diameters, arceostyle^ or columns thin-set. In all the 
orders, when there are four columns in one row, the arrange- 
ment is called, tetrastyle ^ when there are six in a row, hexa- 
style / and when eight, octastyle. 

204. — The Ionic order is appropriate for churches, colleges, 
seminaries, libraries, all edifices dedicated to literature and the 
arts, and all places of peace and tranquillity. The front of the 



94: 



AECHITECrtrRE. 



Merchants' Exchange, New York city, is a good specimen of 
this order. 

205. — To describe the Ionic volute. Draw a perpendicular 
from a to s^ {Fig. 122,) and make a s equal to 20 min. or to -f 
of the whole height, a c ; draw s <9, at right angles to s a^ and 
equal to 1 J min. ; upon 6>, with 2 J min. for radius, describe the 
eye of the volute ; about 6>, the centre of the eye, draw the 
square, r ^ 1 2, with sides equal to half the diameter of the 
eye, viz. 2i min., and divide it into 144 equal parts, as shown 




AMEEICAN HOUSE-CAEPENTEE. 



95 




Fig. 123. 



at Fig. 123. The several centres in rotation are at the angles 
formed by the heavy lines, as figured, 1, <2, 3, 4, 5, 6, &c. The 
position of these angles is determined by commencing at the 
point, 1, and making each heavy line one part less in length 
than the preceding one. JSTo. 1 is the centre for the arc, a 5, 
{Fig. 122 ;) 2 is the centre for the arc, he; and so on to the 
last. The inside spiral line is to be described from the centres, 
a?, ai, a?, &c., {Fig. 123,) being the centre of the first small 
square tov^ards the middle of the eye from the centre for the 
outside arc. The breadth of the fillet at a y, is to be made 
equal to 2y^o- i^iii- This is for a spiral of three revolutions ; but 
one of any number of revolutions, as 4 or 6, may be drawn, by 
dividing of., {Fig. 123,) into a corresponding number of equal 
parts. Then divide the part nearest the centre, 9, into two 
parts, as at A; join o and 1, also o and 2; draw A 3, parallel 
to 1, and A 4, parallel to 6> 2 ; then the lines, (9 1, c> 2, A 3, A 4, 
will determine the length of the heavy lines, and the place of 
the centres. (See AH. 489.) 



96 AMERICAN HOUSE-CARPENTER. 

206. — ^The Corinthian Order, {Fig. 125,) is in general like 
the Ionic, tliongli the proportions are lighter. The Corinthian 
displays a more airy elegance, a richer appearance ; but its 
distinguishing feature is its beautiful capital. This is gene- 
rally supposed to have had its origin in the capitals of the 
columns of Egyptian temples ; which, though not approaching 
it in elegance, have yet a similarity of form with the Corin- 
thian. The oft-repeated story of its origin which is told by 
Yitruvius — an architect who flourished in Rome, in the days 
of Augustus Caesar — though pretty generally considered to be 
fabulous, is nevertheless worthy of being again recited. It is 
this : a young lady of Corinth was sick, and finally died. 
Her nurse gathered into a doep basket, such trinkets and 
keepsakes as the lady had been fond of when alive, and 
placed them upon her grave ; covering the basket with a flat 
stone or tile, that its contents might not be disturbed. The 
basket was placed accidentally upon the stem of an acanthus 
plant, which, shooting forth, enclosed the basket with its foli- 
age ; some of which, reaching the tile, turned gracefully over 
in the form of a volute. 

A celebrated sculptor, Calima- 
chus, saw the basket thus deco- 
rated, and from the hint which it 
suggested, conceived and con- 
structed a capital for a column. 
This was called Corinthian from 
the fact that it was invented and 
^'^- ^"^^^ first made use of at Corinth. 

207. — ^The Corinthian being the gayest, the richest, and 
most lovely of all the orders, it is appropriate for edifices 
which are dedicated to amusement, banqueting and festiv- 
ity — for all places where delicacy, gayety and splendour are 
desirable. 

208. — In addition to the three regular orders of architecture. 




AECHITECTTJRE. 



r^-r^ 




3,i 



-mi 









.2.8., 
35 



^|:: 



Z 




S^ 



75 jg 



CORINTHIAN ORDER. -Fi-'. 125, 

13 



9S AMERICAN HOUSE-CAEPENTER. 

it was sometimes customary among the Greeks — and after 
wards among other nations — to employ representations of the 
human form, instead of columns, to support entablatures ; these 
were called Persians and Caryatides. 

209. — Persians are statues of men, and are so called in 
commemoration of a victory gained over the Persians by Pau- 
sanias. The Persian prisoners were brought to Athens and 
condemned to abject slavery ; and in order to represent them 
in the lowest state of servitude and degradation, the statues 
were loaded with the heaviest entablature, the Doric. 

210. — Caryatides are statues of women dressed in long 
robes after the Asiatic manner. Their origin is as follows. 
In a war between the Greeks and the Caryans, the latter were 
totally vanquished, their male population extinguished, and 
their females carried to Athens. To perpetuate the memory 
of this event, statues of females, having the form and dress of 
the Caryans, were erected, and crowned with the Ionic or Co- 
rinthian entablature. The caryatides were generally formed 
of about the human size, but the persians much larger ; in 
order to produce the greater awe and astonishment in the 
beholder. The entablatures were proportioned to a statue in 
like manner as to a column of the same height. 

211. — These semblances of slavery have been in frequent 
use among moderns as well as ancients ; and as a relief from 
the stateliness and formality of the regular orders, are capable 
of forming a thousand varieties ; yet in a land of liberty such 
marks of human degradation ought not to be perpetuated. 

212. — Roman Styles. Strictly speaking, Rome had no 
architecture of her own — all she possessed was borrowed from 
other nations. Before the Romans exchanged intercourse 
with the Greeks, they possessed some edifices of considerable 
extent and merit, which were erected by architects from Etru- 
ria ; but Rome was principally indebted to Greece for what 
she acquired of the art. Although there is no such thing aa 



ARCHITECTURE. 



99 



«T7jr:< 




fig. 126. 



^JO AMERICAN HOTJSE-CARPENTEK. 

an architecture of Roman invention, yet no nation, perhaps, 
ever was so devoted to the cultivation of the art as the Ro- 
man. Whether we consider the number and extent of their 
structures, or the lavish richness and splendour with which 
they were adorned, we are compelled to yield to them our 
admiration and praise. At one time, under the consuls and 
emperors, Rome employed 400 architects. The public w^orks 
— such as theatres, circuses, baths, aqueducts, &c. — were, in 
extent and grandeur, beyond any thing attempted in modern 
times. Aqueducts were built to convey water from a distance 
of 60 miles or more. In the prosecution of this work, rocks 
and mountains were tunnelled, and valleys bridged. Some of 
the latter descended 200 feet below the level of the water; 
and in passing them the canals were supported by an arcade, 
or succession of arches. Public baths are spoken of as large 
as cities ; being fitted up with numerous conveniences for 
exercise and amusement. Their decorations were most splen- 
did ; indeed, the exuberance of the ornaments alone was offen- 
sive to good taste. So overloaded with enrichments were the 
baths of Diocletian, that on an occasion of public festivity, 
great quantities of sculpture fell from the ceilings and entabla- 
tures, killing many of the people. 

213. — ^The three orders of Greece were introduced into 
Rome in all the richness and elegance of their perfection. 
But the luxurious Romans, not satisfied with the simple ele- 
gance of their refined proportions, sought to improve upon 
them by lavish displays of ornament. They transformed in 
many instances, the true elegance of the Grecian art into a 
gaudy splendour, better suited to their less refined taste. The 
Romans remodelled each of the orders : the Doric, {I^ig. 126,) 
was modified by increasing the heignt of the column to 8 dia- 
meters ; by changing the echinus of the capital for an ovolo, 
or quarter round, and adding an astragal and neck below it ; 
by placing the centre, instead of one edge, of the first triglyph 



ARCFIITECTURE. 



10] 



58 y 



."i: 




:.Tf;: 
H 



25 
.Ml;::. 



= g$iS^ 








f — — >,, 


^mMkm. 


#il§&^&§#lffi^ 


/ 


L, 


tM^'mM^CM^a 


a\2LV>??tismsfcsii5?cssi5)isa; 




Fig. 127. 



102 AMEKICAN HOUSE-CAEPENTEK. 

ovei the centre of the column; and introducing horizontal 
instead of inclined mutules in the cornice, and in some instan 
ces dispensing with them altogether. The Ionic was modified 
by diminishing the size of the volutes, and, in some specimens, 
introducing a new capital in which the volutes were diago- 
nally arranged, ^Fig, 127.) This new capital has been termed 
modern Ionic. The favorite order at Kome and her colonies 
was the Corinthian, {J^ig. 128.) But this order, the Roman 
artists in their search for novelty, subjected to many altera- 
tions — especially in the foliage of its capital. Into the upper 
part of this, they introduced the modified Ionic capital ; thus 
combining the two in one. This change was dignified with 
the importance of an order^ and received the appellation, 
Composite, or Roman : the best specimen of which is found in 
the Arch of Titus, (Fig. 129.) This style was not much used 
among the Romans themselves, and is but slightly appreciated 
now. 

214. — ^The Tuscan Okdee is said to have been introduced to 
the Romans by the Etruscan architects, and to have been the 
only style used in Italy before the introduction of the Grecian 
orders. However this may be, its similarity to the Doric 
order gives strong indications of its having been a rude imita- 
tion of that style : this is very probable, since history informs 
us that the Etruscans held intercourse with the Greeks at a 
remote period. The rudeness of this order prevented its ex- 
tensive use in Italy. All that is known concerning it is from 
Yitruvius — ^no remains of buildings in this style being found 
among ancient ruins. 

215. — For mills, factories, markets, barns, stables, &c., where 
utility and strength are of more importance than beauty, the 
improved modification of this order, called the modern Tuscan, 
{Fig. 130,) will be useful ; and its simplicity recommends it 
where economy is desirable. 

216. — Egyptian Style. The architecture of the ancient 



ARCHITECTURE. 



103 




Fig. 128. 



104 



AMERICAN HOUSE-CAKPENTEB. 




33 X 
3-2- 
23- 
28 



^^■■ 



^ i^ikimm'^^Vd-iS^iMi:!.&i^^^^^^^^s^ 



^^ 






g g Iff IB m n ; 




^M 




Fiff. 129. 



TUSCAlSr ORDER. 



105 




106 



ARCHITECTURE. 



Egyptians -to which that of the ancient Hindoos bears some re* 
semblance — is characterized by boldness of outline, solidity and 
grandeur. The amazing labyrinths and extensive artificial lakes, 
the splendid palaces and gloomy cemeteries, the gigantic pyramids 
and towering (jbelisks, of the Egyptians, were works of immen- 
sity and durability ; and their extensive remains are enduring 
proofs of the enlightened skill of this once-powerful, but long since 
extinct nation. The principal features of the Egyptian Style of 
architecture are — uniformity of plan, never deviating from right 
lines and angles ; thick walls, having the outer surface slightly 
deviating inwardly from the perpendicular ; the whole building 
low ; roof flat, composed of stones reaching in one piece from pier 
to pier, these being supported by enormous columns, very stout in 
proportion to their height ; the shaft sometimes polygonal, having 
no base but with a great variety of handsome capitals, the foliage 
of these being of the palm, lotus and other leaves ; entablatures 
having simply an architrave, crowned with a huge cavetto orna- 
mented with sculpture ; and the intercolumniation very narrow, 
usually li diameters and seldom exceeding 2|. In the remains 
of a temple, the walls were found to be 24 feet thick ; and at the 
gates of Thebes, the walls at the foundation were 50 feet thick 
and perfectly solid. The immense stones of which these, as well 
as Egyptian walls generally, were built, had both their inside and 
outside surfaces faced, and the joints throughout the body of the 
wall as perfectly close as upon the outer surface. For this reason, 
as well as that the buildings generally partake of the pyramidal 
form, arise their great solidity and durability. The dimensions 
and extent of the buildings may be judged from the temple ol 
Jupiter at Thebes, which was 1400 feet long and 300 feet wide— 
exclusive of the porticos, of which there was a great number. 

It is estimated by Mr. Gliddon, U. S. consul in Egypt, that not 
less than 25,000,000 tons of hewn stone wert employed in the 
erection of the Pyramids of Memphis alone,— or enough to con- 
struct 3,000 Bunker-Hill monuments. Some of the blocKs are 40 



EGYPTIAJS" STYLE. 



IC 



H. p. 




Fig. 131. 



10^ ARCHITECTURE. 

feet long, and polished with emery to a surprising degree. It is 
conjectured that the stone for tJiese pyramids was brought, by 
rafts and canals, from a distance of 6 or 7 hundred miles. 

217. — The general appearance of the Egyptian style of archi- 
tecture is that of solemn grandeur — amounting sometimes to 
sepulchral gloom. For this reason it is appropriate for cemete- 
ries, prisons, (fcc. ; and being adopted for these purposes, it is 
gradually gaining favour. 

A great dissimilarity exists in the proportion, form and general 
features of Egyptian columns. In some instances, there is no 
uniformity even in those of the same building, each differing 
from the others either in its shaft or capital. For practical use ♦ 
in this country, Fig. 131 may be taken as a standard of this 
style. The Halls of Justice in Centre-street, New- York city, is 
a building in general accordance with the principles of Egyptian 
architecture. 

Buildings in General. 

218 — That style of architecture is to be preferred in whicu 
utility, stability and regularity, are gracefully blended with gran- 
deur and elegance. But as an arrangement designed for a warm 
country would be inappropriate for a colder climate, it would seem 
that the style of building ought to be modified to suit the wants 
of the people for whom it is designed. High roofs to resist the 
pressure of heavy snows, and arrangements for artificial heat, are 
indispensable in northern climes ; while they would be regarded 
as entirely out of plare in buildings at the equator. 

219. — Among the Greeks, architecture was employed chiefly 
upon their temples and other large buildings ; and the proportions 
of the orders, as determined by them, when executed to such 
large dimensions, have the happiest effect. But when used for 
small buildings,porticos, porches, &c., especially in country-places 
they are rather heavy and clumsy ; in such cases, more slendei 
proportions will be found to produce a better effect. The 



AMERICAN HOUSE-C^: RPENTER. lOJ^ 

English cottage-style is rather more appropriate, and is be(-om- 
ing extensively practised for small buildings in the country. 

220. — Every building should bear an expression suited to its 
destination. If it be intended for national purposes, it should be 
magnificent — grand ; for a private residence, neat and modest ; 
for a banqueting-house, gay and splendid ; for a monument or 
cemetery, gloomy — melancholy ; or, if for a church, majestic and 
graceful. By some it has been said — "somewhat dark and 
gloomy, as being favourable to a devotional state of feeling ;" but 
such impressions can only result from a misapprehension of the 
nature of true devotion. " Her ways are ways of pleasantness, 
and all her paths are peace." The church should rather be a type 
of that brighter world to which it leads. 

221. — However happily the several parts of an edifice may be 
disposed, and however pleasing it may appear as a whole, yet 
much depends upon its site^ as also upon the character and style 
of the structures in its immediate vicinity, and the degree of cul- 
tivation of the adjacent country. A splendid country-seat should 
have the out-houses and fences in the same style with itself, the 
trees and shrubbery neatly trimmed, and the grounds well cul- 
tivated. 

222. — Europeans express surprise that so many houses in this 
country are built of wood. And yet, in a new country, where 
wood is plenty, that this should be so is no cause for wonder. 
Still, the practice should not be encouraged. Buildings erected 
with brick or stone are far preferable to those of wood ; they are 
more durable ; not so liable to injury by fire, nor to need repairs : 
and will be found in the end quite as economical. A wooden 
house is suitable for a temporary residence only ; and those who 
would bequeath a dwelling to their children, will endeavour to 
buif^ with a more durable material. Wooden cornices and gut- 
ters, attached to brick houses, are objectionable — not only on a.c* 
count of their frail nature, but also because they render the build- 
ing liable to destruction by fire. 



no 



AMERICAN HOTJSE-CAEPENTEE. 




AECHITECTURE. Ill 

223. — ^Dwelling houses are built of various dimensions and 
styles, according to their destination ; and to give designs and 
directions for their erection, it is necessary to know their situa- 
tion and object. A dwelling intended for a gardener, would 
require very different dimensions and arrangements from one 
intended for a retired gentleman — with his servants, horses, 
<fec. ; nor would a house designed for the city be appropriate 
for the country. For city houses, arrangements that would be 
convenient for one family might be very inconvenient for two 
or more. Fig. 132, 133, 134, 135, 136, and 137, represent the 
ichnographical projection, or ground-plan, of the floors of an 
ordinary city house, designed to be occupied by one family 
only. Fig. 139 is an elevation, or front-view, of the same 
house : all these plans are drawn at the same scale — which is 
that at the bottom of Fig. 139. 

Fig. 132 is a Plan of the Under-Cellar. 

a, is the coal-vault, 6 by 10 feet. 

5, is the furnace for heating the house. 

c, d, are front and rear areas. 

Fig. 133 is a Plan of the Basement. 

a, is the library, or ordinary dining-room, 15 by 20 feet. 

6, is the kitchen, 15 by 22 feet. 
(?, is the store-room, 6 by 9 feet. 

d, is the pantry, 4 by 7 feet. 

e, is the china closet, 4 by T feet. 
/*, is the servants' water-closet. 

g, is a closet. 

A, is a closet with a dumb-waiter to the first story above. 

i, is an ash closet under the front stoop. 

j, is the kitchen-range. 

Jc, is the sink for washing and drawing water 

I, are wash trays. 



112 



AMERICAN HOUSE-CAHPENTEB. 




FIff. 13B. 

Second Stor j. 



AECHITECTTJEE. 113 

Fig. 134 is a Plan of the First Story. 

a, is the parlor, 15 by 34 feet. 
J)^ is the dining-room, 16 by 23 feet. 
c, is the vestibule. 

€, is the closet containing the dumb-waiter from the basement. 
y, is the closet containing butler's sink. 
^, ^, are closets. 

A, is a closet for hats and cloaks. 
i^j. are front and rear balconies. 

Fig. 135 is the Second Story. 

a, (X, are chambers, 15 by 19 feet. 

Z>, is a bed-room, 7i by 13 feet. 

c, is the bath-room, 7^ by 13 feet. 

d^ d^ are dressing-rooms, 6 by T^- feet. 

6, 6, are closets. 

y, yj are wardrobes: 

^, p', are cupboards. 

Fig. 136 is the Third Story. 
(I, 6i^, are chambers, 15 by 19 feet. 
^, 5, are bed-rooms, 74 by 13 feet, 
c, c, are closets. 

«^, is a linen closet, 5 by 7 feet. 
6, ^, are dressing-closets, 
y, y are wardrobes. 
^, ^, are cupboards. 

Fig. 137 is the Fourth Story. 
a^ a^ are chambers, 14 by 17 feet. 
h, 5, are bed-rooms, 8| by 17 feet, 
c, c, c, are closets. 
d^ is the step-ladder to the roof. 

15 



114 



AMERICAlf HOUSE-CARPENTER. 




AECHITECTUEE. 115 

Fig. 138 is the Section of the House showing the heights of 
the several stories. 

Fig. 139 is the Front Elevation. 

The size of the house is 25 feet front by 55 feet deep ; this 
is about the average depth, although some are extended to 60 
and 65 feet in depth. 

These are introduced to give some general ideas of the prin- 
ciples to be followed in designing city houses. In placing the 
chimneys in the parlours, set the chimney-breasts equi-distant 
from the ends of the room. The basement chimney-breasts 
may be placed nearly in the middle of the side of the room, as 
there is but one flue to pass through the chimney-breast above ; 
but in the second story, as there are two flues, one from the 
basement and one from the parlour, the breast will have to be 
placed nearly perpendicular over the parlour breast, so as to 
receive the flues w^ithin the jambs of the fire-place. As it is 
desirable to have the cliimney-breast as near the middle of the 
room as possible, it may be placed a few inches towards that 
point from over the breast below. So in arranging those of 
the stories above, always make provision for the flues from 
below. 

22tl:. — In placing the stairs, there should be at least as much 
room in the passage at the side of the stairs, as upon thera ; 
and in regard to the length of the passage in the second story, 
there must be room for the doors which open from each of the 
principal rooms into the hall, and more if the stairs require it. 
Having assigned a position for the stairs of the second story, 
now generally placed in the centre of the depth of the house, 
let the winders of the other stories be placed perpendicularly 
over and under them ; and be careful to provide for head- 
room. To ascertain this, when it is doubtful, it is well to draw 
a vertical section of the whole stairs ; but in ordinary cases, 
this is not necessary. To dispose the windows properly, the 



116 



AMERICAN HOUSE-CARPENTEE. 





.U.O- 



fP 



fS 20 2S SO Si 



Fig. 139. 
Aeraticn. 






ARCHITECTURE. 117 

middle window of eacli story should be exactly in the middle 
of the front ; but the pier between the two windows which 
light the parlour, should be in the centre of that room ; be- 
cause when chandeliers or any similar ornaments, hang from 
the centre-pieces of the parlour ceilings, it is important, in 
order to give the better effect, that the pier-glasses at the 
front and rear, be in a range with them. If both these ob- 
jects cannot be attained, an approximation to each must be 
attempted. The piers should in no case be less in width than 
the window openings, else the blinds or shutters when thrown 
open will interfere with one another ; in general practice, it is 
well to make the outside piers § of the width of one of the 
middle piers. When this is desirable, deduct the amount of 
the three openings from the width of the front, and the re- 
mainder will be the amount of the width of all the piers ; 
divide this by 10, and the product will be § of a middle pier ; 
and then, if the parlour arrangements do not interfere, give 
twice this amount to each corner pier, and three times the 
same amount to each of the middle piers. 

PRINCIPLES OF ARCHITECTURE. 

225. — In the construction of the first habitations of men, 
frail and rude as they must have been, the first and principal 
object was, doubtless, utility — a mere shelter from sun and 
rain. But as successive storms shattered the poor tenement, 
man was taught by experience the necessity of building with 
an idea to durability. And when in his walks abroad, the 
symmetry, proportion and beauty of nature met his admiring 
gaze, contrasting so strangely with the misshapen and dispro- 
portioned work of his own hands, he was led to make gradual 
changes ; till his abode was rendered not only commodious 
and durable, but pleasant in its appearance ; and building 
became a fine art, having utility for its basis. 



118 AMERICAN HOTJSE-CARPENTER. 

226. — In all designs for buildings of importance, utility, 
durability and beauty, the first great principles of architec* 
ture, should be pre-eminent. In order that the edifice be 
useful, commodious and comfortable, the arrangement of the 
apartments should be such as to fit them for their several des- 
tinations ; for public assemblies, oratory, state, visitors, retir- 
ing, eating, reading, sleeping, bathing, dressing, &c. — these 
should each have its own peculiar form and situation. To 
accomplish this, and at the same time to make their relative 
situation agreeable and pleasant, producing regularity and 
harmony, require in some instances much skill and sound 
judgment. Convenience and regularity are very important, 
and each should have due attention ; yet when both cannot 
be obtained, the latter should in most cases give place to the 
former. A building that is neither convenient nor regular, 
whatever other good qualities it may possess, will be sure of 
disapprobation. 

227. — ^The utmost importance should be attached to such 
arrangements as are calculated to promote health: among 
these, ventilation is by no means the least. For this purpose, 
the ceilings of the apartments should have a respectable 
height ; and the sky-light, or any part of the roof that can be 
made moveable, should be arranged with cord and pullies, so 
as to be easily raised and lowered. Small openings near the 
ceiling, that may be closed at pleasure, should be made in the 
partitions that separate the rooms from the passages — espe- 
cially for those rooms which are used for sleeping apartments. 
All the apartments should be so arranged as to secui-e their 
being easily kept dry and clean. In dwellings, suitable apart- 
ments should be fitted up for lathing with all the necessary 
apparatus for conveying the water. 

22"^. — To i sure stability in an edifice, it should be designed 
upon well-known geometrical principles : such as science has de- 
monstrated to be necessary and sufficient fcr firmness and dura 



AMERICAN HOUSE-CARPENTER. 119 

bilily. It is well, also, that it have the appearance of stability as 
well as the reality ; for should it seem tottering and unsafe, the 
sensation of fear, rather than those of admiration and pleasure, 
will be excited in the beholder. To secure certainty and accu- 
racy in the application of those principles, a knowledge of the 
strength and other properties of the materials used, is indispensa- 
ble ; and in order that the whole design be so made as to be 
capable of execution, a practical knowledge of the requisite 
mechanical operations is quite important. 

229. — The elegance of an architectural design, although chiefly 
depending upon a just proportion and harmony of the parts, will 
be promoted by the introduction of ornaments — provided this be 
judiciously performed. For enrichments should not only be of a 
proper character to suit the style of the building, but should also 
have their true position, and be bestowed in proper quantity. The 
most common fault, and one which is prominent in Roman archi- 
tecture, is an excess of enrichment : an error which is carefully 
to be guarded against. But those who take the Grecian models 
for their standard, will not be liable to go to that extreme. In 
ornamenting a cornice, or any other assemblage of mouldings, at 
least every alternate member should be left plain ; and those that 
are near the eye should be more finished than those which are dis- 
tant. Although the characteristics of good architecture are utili- 
ty and elegance, in connection with durability, yet some buildings 
are designed expressly for use, and others again for ornament : in 
the former, utility, and in the latter, beauty, should be the gov- 
erning principle. 

230.— The builder should be intimately acquainted with the 
piinciples upon which the essential, elementary parts of a build- 
ing are founded. A scientific knowledge of these will insure 
certainty and security, and enable the mechanic to erect the most 
extensive and lofty edifices with confidence. The more important 
parts are the foundation, the column, the wall, the lintel, the arch, 
the vault, the dome and the roof. A separate description of the 



120 ARCHITECTURE. 

peculiarities of each, would seem to be necessary ; and canno? 
perhaps be better expressed than in the following language of a 
modern writer on this subject. 

231. — "In laying the Foundation of any building, it is ne- 
cessary to dig to a certain depth in the earth, to secure a solid 
basis, below the reach of frost and common accidents. The 
most solid basis is rock, or gravel which has not been moved. 
Next to these are clay and sand, provided no other excavations 
have been made in the immediate neighbourhood. From this 
basis a stone wall is carried up to the surface of the ground, and 
constitutes the foundation. Where it is intended that the super- 
structure shall press unequally, as at its piers, chimneys, or 
columns, it is sv^metimes of use to occupy the space between the 
points of pressure by an inverted arch. This distributes the 
pressure equally, and prevents the foundation from springing be- 
tween the different points. In loose or muddy situations, it is 
always unsafe to build, unless we can reach the solid bottom 
below. In salt marshes and flats, this is done by depositing tim- 
bers, or driving wooden piles into the earth, and raising walls 
upon them. The preservative quality of the salt will keep these 
timbers unimpaired for a great length of time, and makes the 
foundation equally secure with one of brick or stone. 

232. — The simplest member in any building, though by no 
means an essential one to all, is the Column, ox pillar. This is 
a perpendicular part, commonly of equal breadth and thickness, 
not intended for the purpose of enclosure, but simply for the sup- 
port of some part of the superstructure. The principal force 
which a column has to resist, is that of perpendicular pressure. 
In its shape, the shaft of a column should not be exactly cylin- 
drical, but, since the lower part must STipport the weight of the 
superior part, in addition to the weight which presses equally on 
the whole column, the thickness should gradually decrease from 
bottom to top. The outline of columns should be a little curved, 
so as to represent a portion of a very long spheroid, or paraboloid, 



AMERICAN HOUSE-CARPENTER. 121 

rather than of a cone. This figure is the joint result of two cal- 
culations, independent of beauty of appearance. One of these 
is, that the form best adapted for stabiht.y of base is that of a 
cone; the other is, that the figure, which would be of equal 
strength throughout for supporting a superincumbent weight, 
would be generated by the revolution of two parabolas round the 
axis of the column, the vertices of the curves being at its ex- 
tremities. The swell of the shafts of columns was called the en- 
tasis by the ancients. It has been lately found, that the columns 
of the Parthenon, at Athens, which have been commonly sup 
posed straight, deviate about an inch from a straight line, and 
that their greatest swell is at about one third of their height. 
Columns in the antique orders are usually made to diminish one 
sixth or one seventh of their diameter, and sometimes even one 
fourth. The Gothic pillar is commonly of equal thickness 
throughout. 

233. — The Wall, another elementary part of a building, may 
be considered as the lateral continuation of the column, answer- 
ing the purpose both of enclosure and support. A wall must 
diminish as it rises, for the same reasons, and in the same propor- 
tion, as the column. It must diminish still more rapidly if it ex- 
tends through several stories, supporting weights at different 
heights. A wall, to possess the greatest strength, must also con- 
sist of pieces, the upper and lower surfaces of whic^ are horizon- 
tal and regular, not rounded nor oblique. The walls of most of 
the ancient structures which have stood to the present time, are 
constructed in this manner, and frequently have their stones bound 
together with bolts and cramps of iron. The same method is 
adopted in such modern structures as are intended to possess great 
strength and durability, and, in some cases, the stones are even 
dove-tailed together, as in the light-houses at Eddystone and Bell 
Rock, But many of our modern stone walls, for the sake ol 
cheapness, have only one face of the stones squared, the inner 

half of the wall being completed with brick ; so that they can^ 

16 



122 



ARCHITECTURE. 



in reality, be considered only as brick walls faced with stone 
Such walls are said to be liable to become convex outwardly, frora 
the difference in the shrinking of the cement. Ruhhle walls are 
made of rough, irregular stones, laid in mortar. The stones 
should be broken, if possible, so as to produce horizontal surfaces 
The coffer walls of the ancient Romans were made by enclosing 
successive portions of the intended wall in a box, and filling it 
with stones, sand, and mortar, promiscuously. This kind of 
structure must have been extremely insecure. The Pantheon, 
and various other Roman buildings, are surrounded with a double 
brick wall, having its vacancy filled up with loose bricks and 
cement. The whole has gradually consolidated into a mass ot 
great firmness. 

The reticulated walls of the Romans, having bricks with 
oblique surfaces, would, at the present day, be thought highly 
unphilosophical. Indeed, they could not long have stood, had it 
not been for the great strength of their cement. Modern brick 
walls are laid Avith great precision, and depend for firmness more 
upon their position than upon the strength of their cement. The 
bricks being laid in horizontal courses, and continually overlaying 
each other, or breaking joints^ the whole mass is strongly inter- 
woven, and bound together. Wooden walls, composed of timbers 
covered with boards, are a common, but more perishable kind. 
They require to be constantly covered with a coating of a foreign 
substance, as paint or plaster, to preserve them from spontaneous 
decomposition. In some parts of France, and elsewhere, a kind 
of wall is made of earth, rendered compact by ramming it in 
moulds or cases. This method is called building in pise^ and is 
much more durable than the nature of the material would lead 
us to suppose. Walls of all kinds are greatly strengthened by 
angles and curves, also by projections, such as pilasters, chimneys 
and buttresses. These projections serve to increase the breadth 
of the foundation, and are always to be made use of in la^'ge 
buildings, and in walls of considerable length. 



AMERICAN HOUSE-CARPENTER. 123 

234.— The Lintel, oy beam, extends in a right line over a 
vacant space, from one column or wall to another. The strength 
of the lintel will be greater in proportion as its transverse vertical 
diameter exceeds the horiz^yntal, the strength being always as the 
square of the depth. The Jloo?^ is the lateral continuation or. 
connection of beams by means of a covering of boards. 

235. — The Arch is a transverse member of a building, an- 
swering the same purpose as the lintel, but vastly exceeding it in 
strength. The arch, unlike the lintel, may consist of any num- 
ber of constituent pieces, without impairing its strength. It is, 
however, necessary that all the pieces should possess a uniform 
shape, — the shape of a portion of a wedge, — and that the joints, 
formed by the contact of their surfaces, should point towards a 
common centre. In this case, no one portion of the arch can be 
displaced or forced inward ; and the arch cannot be broken by 
any force which is not sufficient to crush the materials of which 
it is made. In arches made of common bricks, the sides of which 
are parallel, any one of the bricks might be forced inward, were 
it not for the adhesion of the cement. Any two of the bricks, 
however, by the disposition of their mortar, cannot collective- 
ly be forced inward. An arch of the proper form, when com- 
plete, is rendered stronger, instead of weaker, by the pressure of 
a considerable weight, provided this pressure be uniform. While 
building, however, it requires to be supported by a centring of 
the shape of its internal surface, until it is complete. The upper 
stone of an arch is called the key-stone, but is not more essential 
than any other. In regard to the shape of the arch, its most 
simple form is that of the semi-circle. It is, however, very fre- 
quently a smaller arc of a circle, and, still more frequently, a por- 
tion of an ellipse. The simplest theory of an arch supporting 
itself only, is that of Dr. Hooke. The arch, when it has only 
its own weight to bear, may be considered as the inversion of a 
chain, suspended at each end. The chain hangs in such a form, 
that the weiglit of each link or portion is held in equilibrium by 



124 ARCHITECTURE. 

the result of two forces acting at its extremities ; and these forces, 
or tensions, are produced, the one by the weight of the portion of 
the chain below the link, the other by the same weight increased 
by that of the link itself, both of them acting originally in a ver- 
tical direction. Now, supposing the chain inverted, so as to con- 
stitute an arch of the same form and weight,' the relative situa- 
tions of the forces will be the same, only they will act in contrary 
directions, so that they are compounded in a similar manner, and 
balance each other on the same conditions. 

The arch thus formed is denominated a catenary arch. In 
common cases, it differs but little from a circular arch of the extent 
of about one third of a whole circle, and rising from the abut- 
ments with an obliquity of about 30 degrees from a perpendicu- 
lar. But though the catenary arch is the best form for support- 
ing its own weight, and also all additional weight which presses 
in a vertical direction, it is not the best form to resist lateral 
pressure, or pressure like that of fluids, acting equally in all direc- 
tions. Thus the arches of bridges and similar structures, when 
covered with loose stones and earth, are pressed sideways, as well 
as vertically, in the same manner as if they supported a weight 
of fluid. In this case, it is necessary that the arch should arise 
more perpendicularly from the abutment, and that its general 
figure should be that of the longitudinal segment of an ellipse. 
In small arches, in common buildings, where the disturbing 
force is not great, it is of little consequence what is the shape ot 
the curve. The outlines may even be perfectly straight, as in the 
tier of bricks which we frequently see over a window. This is, 
strictly speaking, a real arch, provided the surfaces of the bricks 
tend towards a common centre. It is the weakest kind of arch, 
and a part of it is necessarily superfluous, since no greater portion 
can act in supporting a weight above it, than can be included be- 
tween two curved or arched lines. 

Besides the arches already mentioned, various others are in use 
The acute or la ncet arch, much used in Gothic architecture, is 



AMERICAN HOUSE-CARPENTER. 125 

described usually from two centres outside the arch. It is a 
strong arch for supporting vertical pressure. The rampant arch 
is one in which the two ends spring from unequal heights. The 
horse-shoe or Moorish arch is described from one or more centres 
placed above the base line. In this arch, the lower parts are in 
danger of being forced inward. The ogee arch is concavo-con- 
vex, and therefore fit only for ornament. In describing arches^ 
the upper surface is called the extrados^ and the inner, the in- 
trados. The springing lines are those where the intrados meets 
the abutments, or supporting walls. The span is the distance 
from one springing line to the. other. The wedge-shaped stones, 
which form an arch, are sometimes called voussoirs, the upper- 
most being the key-stone. The part of a pier from which ar 
arch springs is called the impost, and the curve formed by the 
upper side of the voussoirs, the archivolt. It is necessary that 
the walls, abutments and piers, on which arches are supported, 
should be so firm as to resist the lateral thrust, as well as vertical 
pressure, of the arch. It will at once be seen, that the lateral or 
sideway pressure of an arch is very considerable, when we recol- 
lect that every stone, or portion of the arch, is a wedge, a part of 
whose force acts to separate the abutments. For want of atten- 
tion to this circumstance, important mistakes have been committed, 
the strength of buildings materially impaired, and their ruin ac- 
celerated. In some cases, the want of lateral firmness in the 
walls is compensated by a bar of iron stretched across the span ot 
the arch, and connecting the abutments, like the tie-beam of a 
roof. This is the case in the cathedral of Milan and some other 
Gothic buildings. 

In an arcade, or continuation of arches, it is only necessary that 
the outer supports of the terminal arches should be strong enough 
to resist horizontal pressure. In the intermediate arches, the lat- 
eral force of each arch is counteracted by the opposing lateral 
force of the one contiguous to it. In bridges, however, where 
individual arches are liable to be destroyed by accident, it is desi 



126 



ARCHITECTURE. 



rable that each of the piers should possess sufficient horizontal 
strength to resist the lateral pressure of the adjoining arches. 

236. — The Vault is the lateral continuation of an arch, serving 
to cover an area or passage, and bearing the same relation to the 
arch that the wall does to the column. A simple vault is con- 
structed on the principles of the arch, and distributes its pressure 
equally along the walls or abutments. A complex or groiJied 
vault is made by two vaults intersecting each other, in which 
case the pressure is thrown upon springing points, and is greatly 
increased at those points. The groined vault is common in 
Gothic architecture. 

237. — The Dome, sometimes called cupola^ is a concave cover- 
ing to a building, or part of it, and may be either a segment of a 
sphere, of a spheroid, or of any similar figure. When built of 
stone, it is a very strong kind of structure, even more so than the 
arch, since the tendency of each part to fall is counteracted, not 
only by those above and below it, but also by those on each side. 
It is only necessary that the constituent pieces should have a 
common form, and that this form should be somewhat like the 
frustum of a pyramid, so that, when placed in its situation, its 
four angles may point toward the centre, or axis, of the dome. 
During the erection of a dome, it is not necessary that it should 
be supported by a centring, until complete, as is done in the arch. 
Each circle of stones, when laid, is capable of supporting itself 
without aid from those above it. It follows that the dome may 
be left open at top, without a key-stone, and yet be perfectly 
secure in this respect, being the reverse of the arch. The dome 
of the Pantheon, at Rome, has been always open at top, and yet 
has stood unimpaired for nearly 2000 years. The upper circle 
of stones, though apparently the weakest, is nevertheless often 
made to support the additional weight of a lantern or tower above 
it. In several of the largest cathedrals, there are two domes, one 
within the other, which contribute their joint support to the lan- 
tern, which rests upon the top. In these buildings, the dome 



AMERICAN HOUSE-CARPENTER. 



127 



rests upon a circular wai], which is supported, in its turn, by 
arches upon massive pillars or piers. This construction is called 
building upon pendenfivesj and gives open space and room for 
passage beneath the dome. The remarks which have been made 
in regard to the abutments of the arch, apply equally to the walls 
immediately supporting a dome. They must be of sufficient 
thickness and solidity to resist the lateral pressure of the dome, 
which is very great. The walls of the Roman Pantheon are of 
great depth and solidity. In order that a dome in itself should be 
perfectly secure, its lower parts must not be too nearly vertical, 
since, in this case, they partake of the nature of perpendicular 
walls, and are acted upon by the spreading force of the parts above 
them. The dome of St. Paul's church, in London, and some 
others of similar construction, are bound with chains or hoops ot 
iron, to prevent them from spreading at bottom. Domes which 
are made of wood depend, in part, for their strength, on their in- 
ternal carpentry. The Halle du Bled, in Paris, had originally a 
wooden dome more than 200 feet in diameter, and only one foot 
in thickness. This has since been replaced by a dome of iion 
(See Art. 389.) 

238. — The Roof is the most common and cheap method of 
covering buildings, to protect them from rain and other effects of 
the weather. It is sometimes flat, but more frequently oblique, in 
its shape. The flat or platform-roof is the least advantageous for 
shedding rain, and is seldom used in northern countries. The 
pent roof, consisting of two oblique sides meeting at top, is the 
most common form. These roofs are made steepest in cold cli- 
mates, where they are liable to be loaded with snow. Where the 
four sides of the roof are all oblique, it is denominated a hipped 
roof, and where there are two portions to the roof, of different ob- 
liquity, it is a curb, or mansard roof. In modern times, roofs 
are made almost exclusively of wood, though frequently covered 
with incombustible materials. The internal structure or carpen- 
try of roofs is a subject of considerable mechanical contrivance, 



128 ARCHITECTURE. 

The roof is supported by rafters^ which abut on the walls on 
each side, like the extremities of an arch. If no other timbers 
existed, except the rafters, they would exert a strong lateral pres- 
jsure on the walls, tending to separate and overthrow them. To 
counteract this lateral force, a tie-heam^ as it is called, extends 
across, receiving the ends of the rafters, and protecting the wall 
from their horizontal thrust. To prevent the tie-beam from 
sagging^ or bending downward with its own weight, a king- 
post is erected from this beam, to the upper angle of the rafters, 
serving to connect the whole, and to suspend the weight of the 
beam. This is called trussing. Queen-jjosts are sometimes 
added, parallel to the king-post, in large roofs ; also various other 
connecting timbers. In Gothic buildings, where the vaults do 
not admit of the use of a tie-beam, the rafters are prevented from 
spreading, as in an arch, by the strength of the buttresses. 

In comparing the lateral pressure of a high roof with that of a 
low one, the length of the tie-beam being the same, it will be 
seen that a high roof, from its containing most materials, may 
produce the greatest pressure, as far as weight is concerned. On 
the other hand, if the weight of both be equal, then the low roof 
will exert the greater pressure ; and this will increase in propor- 
tion to the distance of the point at which perpendiculars, drawn 
from the end of each rafter, would meet. In roofs, as well as in 
wooden domes and bridges, the materials are subjected to an in- 
ternal strain, to resist which, the cohesive strength of the material 
is relied on. On this account, beams should, when possible, be 
of one piece. Where this cannot be effected, two or more beams 
are connected together by splicing. Spliced beams are never so 
strong as whole ones, yet they may be made to approach the same 
strength, by affixing lateral pieces, or by making the ends overlay 
each other, and connecting them with bolts and straps of iron. 
The tendency to separate is also resisted, by letting the two piecea 
Into each Mher by the process called scarfing. Mortices, inr 



AMERICAN HOUSE-CARPENTER. 129 

tended to truss or suspend one piece by another, should be formed 
upon similar principles. 

Roofs in the United States, after being boarded, receive a s .- 
condary covering of shingles. When intended to be incombustib* , 
they are covered with slates orearthern tiles, or with sheets of lead, 
copper or tinned iron. Slates are preferable to tiles, being lightci , 
and absorbing less moisture. Metg,llic sheets are chiefly used for 
flat roofs, wooden domes, and curved and angular surfaces, which 
require a flexible material to cover them, or have not a sufficient 
pitch to shed the rain from slates or shingles. Various artificial 
compositions are occasionally used to cover roofs, the most com- 
mon of which are mixtures of tar with lime, and sometimes with 
sand and gravel." — Ency. Am. (See Art. 354.) 

17 



SECTION III.— MOULDINGS, CORNICES, <fcr 



MOULDINGS. 



239. — A moulding is so called, because of its being ol the 
same determinate shape along its whole length, as though the 
whole of it had been cast in the same mould or form. The regulai 
mouldings, as found in remains of ancient architecture, are eight 
in number ; and are known by the following names : 



I ~1 Annulet, band, cincture, fillet, listel or square. 

Fig. 140. 



Fig. 141. 



J Astragal or bead. 



} 



Torus 01 tore. 



Fig. 142. 



L 



Scotia, trochilus or mouth. 



Fig. 143. 



Ovolo, quarter-round or ecbimis. 

Fig. 144. 



AMERICAN HOUSE-CARPENTER. 



131 




CavettOj cove or hollow. 



Cymatium, or cyma-recta. 



Fig. 147. 



Inverted cymatium, or cyma-reversa 



Ogee. 



Some of the terms are derived thus : fillet, from the French 
wordjil, thread. Astragal, from astragalos, a bone of the heel 
— or the curvature of the heel. Bead, because this moulding, 
when properly carved, resembles a string of beads. Torus, or 
tore, the Greek for rope, which it resembles, when on the base of 
a column. Scotia, from shotia, darkness, because of the strong 
shadow which its depth produces, and which is increased by the 
projection of the torus above it. Ovolo, from ovum, an a^^, 
which this member resembles, when carved, as in the Ionic capi- 
tal. Cavetto, from cavus, hollow. Cymatium, from kumaton 
a wave. 

240. — Neither of these mouldings is peculiar to any one of the 
orders of architecture, but each one is common to all ; and al- 
though each has its appropriate use, yet it is by no means con- 
fined to any certain position in an assemblage of mouldings 
The use of the fillet is to bind the parts, as also that of the astra- 
gal and torus, which resemble ropes. The ovolo and cyma-re- 
versa are strong at their upper extremities, and are therefore used 
to support projecting parts above them. The cyma-recta and 
cavetto, being weak at their upper extremities, are not used as 
supporters, but are placed uppermost to cover and shelter the 
other parts. The scotia is introduced in the base of a column, to 



132 MOULDINGS. CORNICESj &C. 

separate the upper and lower torus, and to produce a pleasing 
variety and relief. The form of the bead, and that of the torus, 
is the same ; the reasons for giving distinct names to them are, 
that the torus, in every order, is always considerably larger than 
the bead, and is placed among the base mouldings, whereas the 
bead is never placed there, but on the capital or entablature ; the 
torus, also, is seldom carved, whereas the bead is; and while the 
torus among the Greeks is frequently elliptical in its form, the 
bead retains its circular shape. While the scotia is the reverse of 
the torus, the cavetto is the reverse of the ovolo, and the cyma- 
recta and cyma-reversa are combinations of the ovolo and cavetto. 

241. — The curves of mouldings, in Roman architecture, were 
most generally composed of parts of circles ; while those of the 
Greeks were almost always elliptical, or of some one of the conic 
sections, but rarely circular, except in the case of the bead, which 
was always, among both Greeks and Romans, of the form of a 
semi-circle. Sections of the cone afford a greater variety ol 
forms than those of the sphere ; and perhaps this is one reason 
why the Grecian architecture so much excels the Roman. The 
quick turnings of the ovolo and cyma-reversa, in particular, when 
exposed to a bright sun, cause those narrow, well-defined streaks 
of light, which give life and splendour to the whole. 

242. — K profile is an assemblage of essential parts and mould- 
ings. That profile produces the happiest effect which is com- 
posed of but few members, varied in form and size, and arranged 
so that the plane and the curved surfaces succeed each other al- 
ternately. 

243. — To describe tke Grecian torus and scotia. Join the 
extremities, a and 6, [Fig. 148;) and from/, the given projection 
of the moulding, draw/o, at right angles to the fillets; from 6, 
draw h A, at right angles to a b ; bisect a b in c; join / ..^.d c, 
and upon c, with the radius, cf, describe the arc, / A, cutting b h 
in h ; through c, draw d e, parallel with the fillets ; make d c and 
c e, each equal to 6 A ; then d e and a b will be conjugate diame 



AMERICAN HOUSE-CARPENTER, 



133 




Fig. 148. 



ters of the required ellipse. To describe the curve by intersec- 
tion of lines, proceed as directed at Art. 118 and note ; by a 
trammel, see Art, 116 ; and to find the foci, in order to describe it 
with a string, see Art, 115. 




d 


\ 






a 



Fig. 149. 



Fig. 150. 



244. — Fig. 149 to 156 exhibit various modifications of the 
Grecian ovolo, sometimes called echinus. Fig. 149 to 153 ar« 



\u 



MOULDINGS, CORNICES, &C. 





F^ J 


c 






I 



Fig. 151. 




Fig. 152. 




> 




Fig. 153. 



Fig. 154. 




Fig. 155. 



Fig. 156. 



elliptical, a h and h c being given tangents to the curve ; parallel 
to which, the semi-conjugate diameters, a d and d c, are drawn. 
In Fig. 149 and 150, the lines, a d and d c, are semi-£ixes, the 
tangents, a h and h c, being at right angles to each other. To 
draw the curve, see Art. 118. In Fig. 153, the curve is para- 
bolical, and is drawn according to Art. 127. In Fig. 155 and 156, 
the curve is hyperbolical, being described according to Art. 128. 
The length of the transverse axis, a b, being taken at pleasure 
in order to flatten the curve, a b should be made short in proper 
tion to a c. 



AMERICAN HOUSE-CARPENTER. 



13i 





Fig. 58 



Fig. 157. 



24:5. — To describe the Grecian cavetto, {I^^g- 157 and 158,) 
having the height and projection given, see Art. 118. 



W 



. 1 


/■ 


\ 


\ 


\ 


\ 




i 





Fig. 159. 



Fig. 160. 



246. — To describe the Grecian cyma-recta. When the pro- 
jection is more than the height, as at Fig. 159, make a b equal 
to the height, and divide abed into 4 equal parallelograms ; 
then proceed as directed in note to Art. 118. When the projec- 
tion is less than the height, draw d a, {Fig. 160,) at right angles 
to a b ; complete the rectangle, abed; divide this into 4 equal 
rectangles, and proceed according to Art. 118. 



1 








^ 


?^ 


r 


a 




■^6 


^ 


s^ 


c 







Fig. 161. 



d 
Fig. 162. 



247. — To describe the Grecian cyma-reversa. Wlien the 



136 



MOULDINGS, CORNICES, &C. 



projection is more than the height, as at Fig. 161, proceed as di 
rected for the last figure ; the curve being the same as that, the 
position only being changed. When the projection is less than 
the height, draw a d, [Fig. 162,) at right angles to the fillet ^ 
make a d equal to the projection of the moulding : then proceed 
as directed for Fig. 159. 

24:8. — Roman mouldings are composed of parts of circles, and 
have, therefore, less beauty of form than the Grecian. The bead 
and torus are of the form of the semi-circle, and the scotia, also, 
in some instances ; but the latter is often composed of two quad- 
rants, having dififerent radii, as at Fig. 163 and 164, which re- 
semble the elliptical curve. The ovolo and cavetto are generally 
a quadrant, but often less. When they are less, as at Fig. 167, 
the centre is found thus : join the extremities, a and 6, and bisect 
ah'm. c ; from c, and at right angles to a b, draw c d, cutting a 
level line drawn from a in d ; then d will be the centre. This 
moulding projects less than its height. When the projection is 
more than the height, as at Fig. 169, extend the line from c until 





Fig. 163. 



Fig. 164. 





Fig. 165. 



Fig. 166. 



AMERICAN HOUSE-CARPENTER. 



137 





g. 168, 



a 




Fig. 169. 



Fig. 170, 



Fig. 171. 











f 


1 



Fig. 172. 




Fig. 173. 




Fig. 174. 



18 



138 



MOULDINGS, CORMCES, &C, 








N 


,^ 


^ 


__J \ 





Fig 175. 



Fig. 176. 




t'ig. 177. 




it cuts a perpendicular drawn from a^ as at d / and that will 
be the centre of the curve. In a similar manner, the centres 
are found for the mouldings at Fig. 164, 168, 170, 173, 174, 
175, and 176. The centres for the curves at Fig. 177 and 178, 
are found thus : bisect the line, <^ 5, at c / upon a^ c and 5, suc- 
cessively, with ^ c or c J for radius, describe arcs intersecting 
at d and d / then those intersections will be the centres. 

249. — Fig. 179 to 186 represent mouldings of modern inven 
tion. They have been quite extensively and successfully used 
in inside finishing. Fig. 179 is appropriate for a b.ed-moulding 
under a low projecting shelf, and is frequently used under man- 
tle-shelves. The tangent, i h, is found thus : bisect the line, a h, 
at (?, and heat d; from d, draw d e, at right angles to eh ; from 
5, draw hf^ parallel to e d ; upon J, with h dior radius, describe 
the arc, df; divide this arc into 7 equal parts, and set one of the 
parts from 5, the limit of the projection, to o / make o h equal to 
oe ; from A, through <?, draw the tangent, h i / divide h A, h c, c i 
and i a^ each into a like number of equal parts, and draw tlie in- 



AMERICAN HOUSE-CARPENTER. 



139 




Fig. 179. 




Jig. 181. 



140 



MOULDINGS, CORNICES, <fcC. 





Pig. 182. 



Fig. 183. 




Fig 184. 



Fie. 185. 



Fig. 186. 



tersecting lines as directed at Art. 89. If a bolder form is 
desired, draw the tangent, i A, nearer horizontal, and describe 
an elliptic curve as shown in Fig. 148 and 181. Fig. 180 is 
much used on base, or skirting of rooms, and in deep panelling. 
The curve is found in the same manner as that of Fig. 179. In 
this case, however, where the moulding has bo little projection 



AMERICAN HOUSE-CARPENTER. 



141 



in comparison with its height, the point, e, being found as in the 
last figure, h s may be made equal to s e, instead of o s as in tne 
last figure. Fig. 181 is appropriate for a crown moulding of a 
cornice. In this figure the height and projection are given ; the 
direction of the diameter, a 6, drawn through the middle cf 
the diagonal, e /, is taken at pleasure ; and d c'ls parallel to a 
€. To find the length of d c, draw b h, at right angles to a b ; 
upon 0, with o f for radius, describe the arc,/ h^ cutting b h in 
h ; then make o c and o d, each equal to b h* To draw the curve, 
see note to Art. 118. Fig. 182 to 186 are peculiarly distinct from 
ancient mouldings, being composed principally of straight lines ; 
the few curves they }X)ssess are quite short and quick 



H. P. 



H. p. 



5 


15 




4 


12i 


n.J 


2 


11 


1 


9 


10* 








10 







li 15 



Hi 



IH 



lOi 



Fig. 187. 



Fig. 188. 



250.— F*^. 187 and 188 are designs for antse caps. The 



* The manner of ascertaining the length of the conjugate diameter, d c, in this figure, 
and also in Fig. 14B, 198 and 199 is new, and is important in this application. It ia 
founded upon well-known mathematical principles, viz : All the parallelograms that may 
be circumscribed about an ellipsis are equal to one another, and consequently any one 
is equal to the rectangle of the two axes. And again : the sum of the squares of every 
pair of conjugate diameters is equal to the sum of the squares of the two axes. 



14:2 



AMEEICAN HOTJSE-CAEPENTER. 



diameter of the antae is divided into 20 equal parts, and the 
height and projection of the members, are regulated in accord- 
ance with those parts, as denoted under IT and JP, height and 
projection. The projection is measured from the middle of 
the antae. These will be found appropriate for porticos, door- 
ways, mantel-pieces, door and window trimmings, &c. The 
height of the antse for mantel-pieces, should be from 5 to 6 
diameters, having an entablature of from 2 to 2J diameters. 
This is a good proportion, it being similar to the Doric order. 
But for a portico these proportions are much too heavy ; an 
antse, 15 diameters high, and an entablature of 3 diameters, 
will have a better appearance. 

CORNICES. 

251. — jFtg. 189 to 197 are designs for eave cornices, and 
I^ig. 198 and 199 are for stucco cornices for the inside finish 
of rooms. In some of these the projection of the uppermost 
member from the facia, is divided into twenty equal parts, 



7 



^ 




Fig 189. 



MOULDINGS, COENICES. &C. 



143 



and the various members are proportioned according^ to those 
parts, as figured under ^and P. 




Fier. 190. 



7 




kiam 



JUJUUUUUUUuUUUUUUU 









Fir. 191. 



144 



AMERICAN HOUSE-CAKPENTEK. 







/ 


7 1™ 


"'T r 


;=3 


d 


^ t 


ST' 


_____ J' 



Fig. 193. 




Fig. 193. 



MOULDINGS, CORNICES, &C. 



145 




Fiff. 194. 



H. P. 








iH 20 

1 

5 

HiTi 

n _ 

8 16: 


I 


1 


V 


J 


J'^ I X 


h^ 


1 1 




> ff ■:, VV-' ■ - 




25 




J' .'/ 





-X 



Fig. 195. 

19 



14f5 



AMERICAN HOUSE-CARPENTER. 



H. P. 



■Ii20 



ill' 

1 



716 



,-'4^ 



i:::^^ 



Fig. 196. 



H.P . 

3i 20 



3i 16 

1' 
4 








Fig. 197. 



IMOULHINGS, CORNICES, &C, 



147 



H. P 



5< 20 

Si 



173 




2i 



lK^SDKDTOOOOOOOO 



Fig. 198. 




Fig. 199. 



1-48 



AMERICAN HOUSE-CARPENTER. 

d ^ 




h 12 3 4c 

Fig. 200. 



252. — To proj)ortio7i an eave cornice in accordance with the 
height of the building. Draw the line, a c, [Fig, 200,) and 
make h c and h a, each equal to 36 inches ; from 6, draw h c?, at 
right angles to a c, and equal in length to 3 of a c ; bisect h dm 
e, and from a, through e, draw a f; upon a, with a c for radius, 
describe the arc. c/, and upon e, with effor radius, describe the 
arc,/c?; divide the curve, df c, into 7 equal parts, as at 10, 20, 
30, &c., and from these points of division, draw lines to b c, pa- 
rallel to d b ; then the distance, b 1, is the projection of a cornice 
for a building 10 feet high ; b 2, the projection at 20 feet high ; 
b 3, the projection at 30 feet, &c. If the projection of a cornice for 
a building 34 feet high, is required, divide the arc between 30 and 
40 into 10 equal parts, and from the fourth point from 30, draw a 
line to the base, b c, parallel with b d ; then the distance of the 
point, at which that line cuts the base, from 6, will be the projec- 
tion required. So proceed for a cornice of any height within 70 
feet. The above is based on the supposition that 36 inches is the 
proper projection for a cornice 70 feet high. This, for general 
purposes, will be found correct; still, the length of the line, b c, 
may be varied to suit the judgment of those who think differ- 
ently. 

Having obtained the projection of a cornice, divide it into 20 
equal parts, and apportion the several members according to its 
destination — as is shown at Fig. 196, 196, and 197. 



MOULDINGS, CORNICES, &C. 
h 



149 



. ./■- 


J ,/ 


/., 


rr 


a^ 


^JJ 



Fig. 201. 

25^1- -To ^proportion a coriiice according to a smaller given 
one. Let the cornice at Fig. 201 be the given one. Upon any 
point in the lowest line of the lowest member, as at a, with the 
height of the required cornice for radius, describe an intersecting 
arc across the uppermost line, as at b ; join a and h : then b 1 will 
DC the perpendicular height of the upper fillet for the proposed cor- 
nice, 1 2 the height of the crown moulding — and so of all the 
members requiring to be enlarged to the sizes indicated on this 
line. For the projection of the proposed cornice, draw a d, at right 
angles to a b, and c d, at right angles to be; parallel with c d, 
draw lines from each projection of the given cornice to the line, 
a c?; then e (i will be the required projection for the proposed 
cornice, and the perpendicular lines falling upon e d will indicate 
the proper projection for the members. 

254. — To proportion a cornice accordifig to a larger given 
one. Let A, {Fig. 202,) be the given cornice. Extend a o to 6, 
and draw c d, at right angles to ab ; extend the horizontal lines 
of the cornice. A, until they touch o d ; place the height of the 
proposed cornice from o to e, and join / and e ; upon o, with the 
projection of the given cornice, o a, for radius, describe the quad- 
rant, ad; from d, draw d b, parallel to/ e ; upon o, with o b for 
radius, describe the quadrant, be; then o c will be the proper pro- 
jection for the proposed cornice. Join a and c ; draw lines from the 



150 



AMERICAN HOUSE-CARPENTER. 




projection of the different members of the given cornice to a o^ 
parallel to o d ; from these divisions on the line, a o, draw lines 
to the line, o c, parallel to a c ; from the divisions on the line, o f. 
draw lines to the line, o e, parallel to the line, f e ; then the di- 
visions on the lines, o e and o c, will indicate the proper height and 
projection for the different members of the proposed cornice. In' 
this process, we have assumed the height, o e, of the proposed 
cornice to be given ; but if the projection, o c, alone be given, we 
can obtain the same result by a different process. Thus : upon o. 
with c for radius, describe the quadrant, c b ; upon o, with o a 
for radius, describe the quadrant, ad; join d and b ; from/, draw 
f e^ parallel to d b ; then o e will be the proper height for the pro- 
posed cornice, and the height and projection of the different mem- 
bers can be obtained by the above directions. By this problem, 
a cornice can be proportioned according to a smaller given one 
as well as to a larger ; but the method described in the previous 
article is much more simple for that purpose. 

255. — To find the angle-bracket for a cornice. Let J., {Fig. 
203,) be the wall of the building, and B the given bracket, which, 
for the present purpose, is turned down horizontally. The angle- 
bracket, C, is obtained thus : through the extremity, a, and paral- 



MOULDINGS, CORNICES, &C. 



151 




g Fig. 203. 



Fiff. 204. 



lei with the wall,/eZ, draw the line, ah ; make e c equal a /, 
and through c, drav/ c h, parallel with e d ; join d and b, and from 
the several angular points in i>', draw ordinates to cut d b inl, 2 
and 3 ; at those points erect lines perpendicular to d b ; from A, 
draw h g, parallel to/ a ; take the ordinates, 1 o, 2 o, &c., at jB, 
and transfer them to C, and the angle-bracket, C, will be defined. 
In the same mailner, the angle-bracket for an internal cornice, or 
the angle-rib of a coved ceiling, or of groins, as at Fig. 204, can 
be found. 

256. — A level crown moulding being given, to find theraking 
Wioulding and a level return at the tojp. Let A, [Fig. 205,) be 
the given moulding, and A b the rake of the roof. Divide the 
curve of the given moulding into any number of parts, equal or 
unequal, as at 1, 2, and 3 ; from these points, draw horizontal 
lines to a perpendicular erected from c; at any convenient place 
on the rake, as at B, draw a c, at right angles to Ab ; also, from 
6, draw the horizontal line, b a ; place the thickness, d a^ of the 
moulding at A, from b to a, and from a, draw the perpendicular 
line, a e ; from the points, 1, 2, 3, at J., draw lines to C, parallel 
io Ab ; make a 1, « 2 and a 3, at B and at C, equal to a 1, (fee, 
at A ; through the points, 1, 2 and 3, at B, trace the curve — this 
will be the proper form for the raking moulding. From 1, 2 and 



152 



AMERICAN HOUSE-CARPENTER. 




Fig 205. 



% at C, drop perpendiculars to the corresponding ordinates from 
1, 2 and 3, at A ; through the poir ts of intersection, trace the 
curve — this will be the proper form lor the return at the top. 



S ECTION IV.— 1 RAMING. 



257. — This subject is, to the carpenter, of the highest impor- 
tance ; and deserves more attention and a larger place in a volume 
oi this kind, than is generally allotted to it. Something, indeed, 
has been said upon the geometrical principles, by which the seve- 
ral lines for the joints and the lengths of timber, may be ascer- 
tained ; yet, besides this, there is much to be learned. For how- 
ever precise or workmanlike the joints may be made, what will 
it avail, should the system of framing, from an erroneous position 
of its timbers, &c., change its form, or become incapable of sus- 
taining even its own weight ? Hence the necessity for a know- 
ledge of the laws of pressure and the strength of timber. These 
bemg once understood, we can with confidence determine the best 
position and dimensions for the several timbers which compose a 
floor or a roof, a partition or a bridge. As systems of framing 
are more or less exposed to heavy weights and strains, and, in 
case of failure, cause not only a loss of labour and material, but 
frequently that of life itself, it is very important that the materials 
employed be of the proper quantity and quality to serve their des 
tination. And, on the other hand, any superfluous material is not 
only useless, but a positive injury, it being an unnecessary load 
upon the points of support. It is necessary, therefore, to know 

20 



154: 



AMERICAN HOUSE-CARPEKTER. 



the least quantity of timber that will suffice for strength. Tho 
greatest fault in framing is that of using an excess of material. 
Economy, at least, would seem to require that this evil be abated. 

Before proceeding to considei the principles upon which a sys- 
tem of framing should be constructed, let us attend to a few of 
the elementary laws in Mechanics^ which will be found to be of 
great value in determining those principles. 

258. — Laws of Pressure. (1.) A heavy body always 
exerts a pressure equal to its own weight in a vertical direction. 
Example: Suppose an iron ball, weighing iUO lbs., be supported 
upon the top of a perpendicular post, [Fig. 220;) then the 
pressure exerted upon that post will be equal to the weight of the 
hall; viz., 100 lbs. (2.) But if two inclined posts, (Fi^. 206,) 
be substituted for the perpendicular support, the united pressures 
upon these posts will be more than equal to the weight, and will 
be in proportion to their position. The farther apart their feet are 
spread the greater will be the pressure, and vice versa. Hence 
tremendous strains may be exerted by a comparatively small 
v.^eight. And it follows, therefore, that a piece of timber intend- 
ed for a strut or post, should be so placed that its axis may coin- 
cide, as near as possible, with the direction of the pressure. The 
direction of the pressure of the weight, W, {Fig. 206^) is in the 
vertical line, h d ; and the weight, W, would fall in that line, if 
the two posts were removed, hence the best position for a support 



•\v 




FRAMING. 



155 



foi- the weiglit would be in that line. But, as it rarely occurs 
in sj^stems of framing that weights can be supported by any 
single resistance, they requiring generally two or more sup* 
ports, (as in the case of a roof supported by its rafters,) it be 
conies important, therefore, to know the exact amount of pres- 
sure any certain weight is capable of exerting upon oblique 
supports. I^ow it has been ascertained that the three hues of 
a triangle, drawn parallel with the direction of three concur- 
ring forces in equilibrium, are in proportion respectively to 
these forces. For example, in Fig. 206, we have a represen- 
tation of three forces concurring in a point, which forces are 
in equilibrium and at rest ; thus, the weight, TT, is one force, 
and the resistance exerted by the two pieces of timber are the 
other two forces. The direction in which the first force acts is 
vertical — downwards ; the direction of the two other forces is 
in the axis of each piece of timber respectively. These three 
forces all tend towards the point, h. 

Draw the axes, a 1) and h c, of the two supports ; make h d 
vertical, and from d draw d e and d f parallel with the axes, 
h G and h a^ respectively. Then the triangle, h d e^ has its 
lines parallel respectively with the direction of the three 
forces ; thus, h d h \n the direction of the weight, W, d e 
parallel wath the axis of the timber h <?, and ^ Z> is in the 
direction of tha timber a 1). In accordance with the principle 
above stated, the lengths of the sides of the triangle, h d e.^ are 
in proportion respectively to the three forces aforesaid ; thus — 

As the length of the line, h d^ 

Is to the number of pounds in the weight, TF", 

So is the length of the line, 1) e^ 

To the number of pounds' pressure resisted by the timber, 
a h. 
Again — 

As the length of the line, h d, 

Is to the number of pounds in the weight, TT, 



156 AlVIEEICAN HOUSE-CAEPENTEE. 

So is the length of the Hne, d e^ 

To the number of pounds' pressure resisted by the timber^ 
he. 
-And again — 

As the length of the line, h e, 
Is to the pounds' pressure resisted by a 5, 
So is the length of the line, d e^ 
To the pounds' pressure resisted by h g. 
These proportions are more briefly stated thus — 
1st Id : W::be: F, 

P being used as a symbol to represent the number of founds' 
pressure resisted by the timber, a h. 

2nd. hd : W:: del Q, 

Q representing the number of pounds' pressure resisted by the 
timber, h e, 

Zd. h e: P :: de: Q. 

259. — ^This relation between lines and pressures is important, 
and is of extensive application in. ascertaining the pressures 
induced by known weights throughout any system, of framing. 
The parallelogram, h e d f^ \q called the Parallelogram of 
Forces / the two lines, h e and h y, being called the convpo- 
nents^ and the line h d the resultant. Where it is required to 
find the components from a given resultant, (Fig. 206,) it is 
not needed to draw the fourth line, df., for the triangle, 1) d e^ 
gives the desired result. But when the resultant is to be 
ascertained from given components, [Fig. 212,) it is more con- 
venient to draw the fourth line. 

260. — Tke Resolution of Forces is the finding of two or 
more forces, which, acting in different directions, shall exactly 
balance the pressure of any given single force. To make a 
practical application of this, let it be required to ascertain 
the oblique pressure in Fig. 206. In this Fig. the line h d 
measures half an inch, (0*5 inch,) and the line h e three- 
tenths of an inch, (0*3 inch.) Now if the weight, TF", be sup- 



FEAMING. 



157 



posed to be 1200 ponncis, then the first stated proportion 
above, 

Id : TT:: le\P, 
becomes 

0-5 : 1200 :: 0-3 : P. 
And since the product of the means divided by one of the 
extremes gives the other extreme, this proportion may be put 
in the form of an equation^ thus — 
1200 X 0-3 



0-5 



P. 



Performing the arithmetical operation here indicated, that is, 
multiplying together the two quantities above the line, and 
dividing the product by the quantity under the line, the quo- 
tient will be equal to the quantity represented by P^ viz,, the 
pressure resisted by the timber, a h. Thus — 

1200 
0-3 



0-5)360-0 



720 r= p. 
The strain upon the timber, a 5, is, therefore, equal to 720 
pounds ; and the strain upon the other timber, 5 c , is also 720 
pounds ; for in this case, the two timbers being inclined 
equally from the vertical, the line e di^ therefore equal to the 
line h e. 




Fig. 207. 



15S AMERICAN HOUSE-CARPENTEE. 

261. — In Fig. 207, the two supports are inclined at different 
angles, and the pressures are proportionately unequal. The 
supports are also unequal in length. The length of the sup- 
ports does not alter the amount of pressure from the concen- 
trated load supported ; but generally long timbers are not so 
capable of resistance as shorter ones. They yield more readily 
laterally, as they are not so stiff, and shorten more^as the com- 
pression is in proportion to the length. To ascertain the pres- 
sures in Fig. 207, let the weight suspended from 5 c? be equal 
to two and three-quarter tons, (2*75 tons.) The line h d mea- 
sures ^YQ and a half tenths of an inch, (0-55 inch,) and the line 
h 6 half an inch, (0*5 inch.) Therefore, the proportion 
hd I W ::1) e: P, becomes 0-55 : 2*75 :: 0-5 : P, 

-, 2-75 X 0-5 ^ 

and — P. 

0-55 

2-75 
0-5 



0-55)l-375(2-5 = P. 
110 



275 
275 



The strain upon the timber, h e^ is, therefore, equal to two 
and a half tons. 

Again, the line e d measures four-tenths of an inch, (0*4 
inch ;) therefore, the proportion 

Id \ W :: ed: Q, becomes 0-55 : 2-75 :: 04 : Q, 
, 2-75 X 04 " 

2-75 
04 

0-55)l-100(2 = Q, 
110 



FEAMING. 159 

The strain upon the timber, 1) f^ is, therefore, equal to two 
ions. 

262. — Thus it is seen that the united pressures exerted by a 
weight upon two inclined supports always exceed the weight. 
In the last case 21 tons exerts a pressure of 2i and two tons, 
equal together to 4-J tons ; and in the former case, 1200 
pounds exerts a pressure of twice Y20 pounds, equal to 1440 
pounds. The smaller the angle of inclination to the horizon- 
tal, the greater will be the pressure upon the supports. So, in 
the frame of a roof, the strain upon the rafters decreases gra- 
dually with the increase of the angle of inclination to the 
horizon, the length of the raftgr remaining the same. 

263. — This is true in comparing systems of framing with 
each other; but in a system where the concentrated weight 
to be supported is not in the middle, (see Fig. 207,) and, in 
consequence, the supports are not inclined equally, the strain 
will be greatest upon the support that has the greatest inclina- 
i'nri to the horizon. 

%^A. — In ordinary cases, in roofs for example, the load is 
aot concentrated but is that of the framing itself. Here the 
amount of the load will be in proportion to the length of the 
rafter, and the rafter increases in length with the increase of 
the angle of inclination, the span remaining the same. So it 
is seen that in enlarging the angle of inclination to the horizon 
in order to lessen the oblique thrust, the load is increased in 
consequence of the elongation of the rafter, thus increasing the 
oblique thrust. Hence there is a limit to the angle of inclina- 
tion. A rafter will have the least oblique thrust when its 
angle of inclination to the horizon is 35° 16^ nearly. This 
angle is attained very nearly when the rafter rises 8-|- inches 
per foot ; or, when the height, B (7, {Fig- 216,) is to the base, 
A C, as 8^ is to 12, or as 0-7071 is to 1-0. 

265. — Correct ideas of the comparative pressures exerted 
mpon timbers, according to their position, will be readily 



160 AMERICAN HOUSE-UAEPENTEE. 

formed by drawing various designs of framing, and estimating 
the several strains in accordance with the parallelogram of 
forces, always drawing the triangle, h d e^ so that the three 
lines shall be parallel with the three forces, or pressures, re- 
sjDectively. The length of the lines forming this triangle is 
unimportant, but it will be found more convenient if the line 
drawn parallel v^ith the known force is made to contain aa 
many inches as the known force contains pounds, or as many 
tenths of an inch as pounds, or as many inches as tons, or 
tenths of an inch as tons : or, in general, as many divisions of 
any convenient scale as there are units of weight or pressure 
in the known force. If drawn in this manner, then the num- 
ber of divisions of the same scale found in the other two lines 
of the triangle will equal the units of pressure or weight of the 
other two forces respectively, and the pressures sought will be 
ascertained simply by applying the scale to the lines of the 
triangle. 

For example, in Fig. 207, the vertical line, 5 d^ of the tri- 
angle, measures fifty-five hundredths of an inch, (0*55 inch ;) 
the line, h e^ fifty-hundred ths, (0*50 inch ;) and the line, e d^ 
forty, (040 inch.) Now, if it be supposed that the vertical pres- 
sure, or the weight suspended below l d, is equal to 55 pounds, 
then the pressure onh e will equal 50 pounds, and that on ^ <^ 
will equal 40 pounds ; for, by the proportion above stated. 
Id iWy.heiP, 
55 : 55 :: 50 : 50 ; 
and so of the other pressure. 

266. — ^If a scale cannot be had of equal proportions with the 
forces, the arithmetical process will be shortened somewhat by 
making the line of the triangle that represents the Jcnown 
weight equal to unity of a decimally divided scale, then the ~ 
other lines will be measured in tenths or hundredths ; and in 
the numerical statement of the proportions between the lines 
and forces, the first term being unity, the fourth term will bo 



FEAMmG. 



161 



ascertained simply by multiplying the second and third terms 
together. 

For example, if the three lines are 1, 0*7 and 1*3, and the 
known weight is 6 tons, then 

h d : W :: h e : P, becomes 
1 : 6 :: 0-r : P = 4-2, 
equals four and two-tenths tons. Again — 

h d : W :: e d : Q, becomes 
1 : 6 :: 1-3 : ^ = Y-8, 
equals seven and eight-tenths tons , 




A 


re 7 


i_ 


a 




G ^ 


t 




i 






i 




A 




6 




u.. 



Fig. 208. 



267. — In Fig. 208 the weight, TT, exerts a pressure on the 
struts in the direction of their length ; their feet, n n^ have, 
therefore, a tendency to move in the direction n <?, and would 
so move, were they not opposed by a sufficient resistance from 
the blocks, A and A. If a piece of each block be cut off at 
the horizontal line, a n, the feet of the struts would slide away 
from each other along that line, in the direction, n a y* but if, 
instead of these, two pieces were cut off at the vertical line, 
n 5, then the struts would descend vertically. To estimate the 
horizontal and the vertical pressures exerted by the struts, let 
n o\)Q made equal (upon any scale of equal parts) to the num- 

21 



162 



AMERICAN HOUSE-CARPENTEE. 



ber of tons with which the strut is pressed ; construct the 
parallelogram of forces by drawing o e parallel to a n, and 0/ 
parallel to h n ; then nf, (by the same scale,) shows the num- 
ber of tons pressure that is exerted by the strut in the direc- 
tion n a, and n e shows the amount exerted in the direction 
n h. By constructing designs similar to this, giving various 
and dissimilar positions to the struts, and then estimating the 
pressures, it will be found in every case that the horizontal 
pressure of one strut is exactly equal to that of the other, how- 
ever much one §trut may be inclined more than the other ; 
and also, that the united vertical pressure of the two struts is 
exactly equal to the weight, W. (In this calculation the 
weight of the timbers has not been taken into consideration, 
simply to avoid complication to the learner. In practice it is 
requisite to include the weight of the framing with the load 
upon the framing.) 




Fig. 209. 



268. — Suppose that the two struts, B and B^ {Fig. 208.) 
were rafters of a roof, and that instead of the blocks, A and A^ 
the walls of a building were the supports: then, to prevent 
the walls from being thrown over by the thrust of B and B, 
it would be desirable to remove the horizontal pressure. Thia 



FBAMING, 



163 



may be done by uniting the feet of the rafters with a rope, 
iron rod, or piece of timber, as in Fig. 209. This figure is 
similar to the truss of a roof. The horizontal strains on the 
tie-beam, tending to pull it asunder in the direction of its 
length, may be measured at the foot of the rafter, as was 
shown at Fig. 208 ; but it can be more readily and as accu- 
rately measured, by drawing from f and e horizontal lines to 
the vertical line, h d^ meeting it in o and o / then f o will be 
the horizontal thrnst at B^ and e o 2X A j these will be fonnd 
to equal one another. When the rafters of a roof are thug 
connected, all tendency to thrust the walls horizontally is 
removed, the only pressure on them is in a vertical direction, 
being equal to the weight of the roof and whatever it has to 
support. This pressure is beneficial rather than otherwise, as 
a roof having trusses thus formed, and the trusses well braced 
to each other, tends to steady the walls. 




Fig. 211. 



269.— i^^>. 210 and 211 exhibit methods of framing for sup- 
porting the equal weights, W and TT. Suppose it be required 



164 AMEEICAiq- HOTJSE-CARPENTEE. 

to measure and compare the strains produced on the pieces, 
A B and A C. Construct the parallelogram of forces, e Ifd^ 
according to Art. 258. Then 5/ will show the strain on A B, 
and 5 e the strain on A 0. By comparing the figures, 5 d be- 
ing equal in each, it will be seen that the strains in Fig. 210 
are about three times as great as those in Fig. 211 : the posi- 
tion of the pieces, A B and A (7, in Fig. 211, is therefore far 
preferable. 




C Fig. 212. 

270. — The Composition of Forces consists in ascertaining the 
direction and amount of one force, which shall be just capable 
of balancing two or more given forces, acting in different 
directions. This is only the reverse of the resolution of forces, 
and the two are founded on one and the same principle, and 
may be solved in the same manner. For example, let A and 
B^ {Fig, 212,) be two pieces of timber, pressed in the direction 
of their length towards 5 — A by a force equal to 6 tons weight, 
and B equal to 9. To find the direction and amount of pres- 
sure they would unitedly exert, draw the lines, h e and 5/*, in 
a line with the axes of the timbers, and make h e equal to the 
pressure exerted by B^ viz., 9 ; also make 1) f equal to the 
pressure on A^ viz., 6, and complete the parallelogram of 
forces, e h f d ; then h d^ the diagonal of the parallelogram, 
will be the direction, and its length, 9*25, will be the amount^ 



FRAMING. 



165 



of the united pressures of A and of B. The line, h d, is 
termed the resultant of the two forces, 5/" and he. If A and 
B are to be supported by one post, G, the best position for 
that post will be in the direction of the diagonal, h d ; and it 
will require to be sufficiently strong to support the united 
pressures of ^ and of ^,wliicli are equal to 9*25 or OJ tons. 




Fig. 213. 



271. — Another example : let JP'ig. 213 represent a piece of 
framing commonly called a crane, which is used for hoisting 
heavy weights by means of the rope, B hf, which passes over 
a pulley at h. This is similar to I^ig. 210 and 211, yet it is 
materially different. In those figures, the strain is in one 
direction only, viz., from h to d / but in this there are two 
strains, from ^ to ^ and from A to W. The strain in the 
direction ^ ^ is evidently equal to that in the direction A W, 
To ascertain the best position for the strut, A O, make 5 e 
equal to hf, and complete the parallelogram of forces, e hfd; 
then draw the diagonal, h d^ and it will be the position re- 
quired. Should the foot, C^ of the strut be placed either 
higher or lower, the strain on ^4. (7 would be increased. In 
constructing cranes, it is advisable, in order that the piece, 
B A, may be under a gentle pressure, to place the foot of the 



166 



AMERICAN HOUSE-CAEPENTEE. 



strut a trifle lower than where the diagonal, h d^ would indi 



cate, but never higher. 



'^A/^^<-JJ 



d 





B 


1. 

Be 












Sa^ 




V y 


LvvvV 



Fig. 214. 



VV 



272. — Ties and Struts. Timbei-vS in a state of tension are 
called ties^ while such as are in a state of compression are 
termed struts. This subject can be illustrated in the following 
manner : 

Let A and B, {Fig. 214,) represent beams of timber support- 
ing the weights, Tf", W and W / A having but one support, 
which is in the middle of its length, and jB two, one at each 
end. To show the nature of the strains, let each beam be 
sawed in the middle from a to h. The effects are obvious : 
the cut in the beam. A, will open, whereas that in B will 
close. If the weights are heavy enough, the beam, A^ will 
break at h ; while the cut in JB will be closed perfectly tight 
at a, and the beam be very little injured by it. But if, on the 
other hand, the cuts be made in the bottom edge of the tim- 
bers, from G to h, B will be seriously injured, while A will 
scarcely be affected. By this it appears evident that, in a 
j)iece of timber subject to a pressure across the direction of its 
length, the fibres are exposed to contrary strains. If the tim- 
ber is supported at both ends, as at B, those from the top edge 
down to the middle are compressed in the direction of their 
length, while those from the middle to the bottom edge are in 
a state of tension ; but if the beam is supported as at A, the 
contrary effect is produced ; while the fibres at the middle of 
either beam are not at all strained. The strains in a framed 



FRAMING. 167 

truss are of the same nature as those in a single beam. The 
truss for a roof, being supported at each end, has its tie-beam 
in a state of tension, while its rafters are compressed in the 
direction of their length. By this, it appears highly important 
that pieces in a state of tension should be distinguished from 
such as are compressed, in order that the former may be pre- 
served continuous. A strut may be constructed of two or 
more pieces ; yet, where there are many joints, it will not 
resist compression so w^ell. 

2Y3. — To distinguish ties from struts. This may be done 
by the following rule. In Fig. 206, the timbers, a h and h c, 
are the sustaining forces, and the weight, W^ is the straining 
force ; and, if the support be removed, the straining force 
would move from the point of support, 5, towards d. Let it be 
required to ascertain whether the sustaining forces are stretched 
or pressed by the straining force. Bute : upon the direction 
of the straining force, & (^, as a diagonal, construct a parallelo- 
gram, e l)fd., whose sides shall be parallel with the direction 
of the sustaining forces, a h and g d '^ through the point, 5, 
draw a line, parallel to tlie diagonal, ef; this may then be 
called the dividing line between ties and struts. Because all 
those supports which are on that side of the dividing line, 
which the straining force would occupy if unresisted, are com- 
pressed, while those on the other side of the dividing line are 
stretched. 

In Fig. 206, the supports are both compressed, being on 
that side of the dividing line which the straining force would 
occupy if unresisted. In Fig. 210 and 211, in which A B and 
A G are the sustaining forces, J. (7 is compressed, 'whereas 
^ ^ is in a state of tension ; A C being on that side of the 
line, h i, which the straining force would occupy if unresisted, 
and A JB on the opposite side. The place of the latter might 
be supplied by a chain or rope. In Fig. 209, the foot of the 
rafter at A is sustained by two forces, the wall and the tie- 



168 



AMERICAN HOITSE-CAEPENTEE. 



beam, one perpendicular and the other horizontal : the direc- 
tion of the straining force is indicated by the line, h a. The 
dividing line, h i, ascertained by the rule, shows that the wall 
is pressed and the tie-beam stretched. 




Fig. 215. 

274.— Another example : let FA B F, {Fig. 215,) represent 
a gate, supported by hinges at A and E. In this case, the 
straining force is the weight of the materials, and the direction 
of course vertical. Ascertain the dividing line at the several 
points, G, B^ I.) J^ ^ and F. It will then appear that the 
force at G is sustained by ^ 6^ and G F^ and the dividing 
line shows that the former is stretched and the latter com- 
pressed. The force at ^is supported by A ^and H F- — the 
former stretched and the latter compressed. The force at B 
is opposed by H B and A B, one pressed, the other stretched. 
The force at F is sustained by G F and F F, G F being 
stretched and i^^ pressed. By this it appears that J. ^ is in 
a state of tension, and F F^ of compression ; also, that A H 
and G F are stretched, while B H and G F are compressed ; 
which shows the necessity of having A H and G F, each in 
one whole length, while B H and G F may be, as they are 
shown, each in two pieces. The force at J is sustained by 
G e/and J 11^ the former stretched and the latter compressed. 



FKAMING. 16S 

The piece, (7-Z>, is neither stretched nor pressed, and conll be 
dispensed with if the joinings at J and / could be made aa 
effectually without it. In case A B should fail, then G D 
would be in a state of tension. 

275. — The centre of gravity. The centre of gravity of a 
uniform prism or cylinder, is in its axis, at the middle of ita 
length ; that of a triangle, is in a line drawn from one angle to 
the middle of the opposite side and at one-third of the length 
of the line from that side ; that of a right-angled triangle, at a 
"Doint distant from the perpendicular equal to one-third of the 
base, and distant from the base equal to one-third of the per- 
pendicular ; that of a pyramid or cone, in the axis and at one- 
quarter of the height from the base. 

276. — The centre of gravity of a trapezoid, (a four-sided 
figure having only two of its sides parallel,) is in a line joining 
the centres of the two ]3arallel sides, and at a distance from 
the longest of the parallel sides equal to the product of the 
length into the sum of twice the shorter added to the longer 
of the parallel sides, divided by three times the sum of the 
two parallel sides. Algebraically thus — 

where d equals the distance from the longest of the parallel 
Bides, I the length of the line joining the two parallel sides, 
and a the shorter and h the longer of the parallel sides. 

JExample. — A rafter, 25 feet long, has the larger end 14 
inches wide, and the smaller end 10 inches wide, how far from 
the larger end is the centre of gravity located ? 

Here, Z = 25, ^ = }f , and h — }f , 

Ij^j^V) _ 25 (2 X If H- 1 1) _ 25 X f I 
hence 6^- 3^^^^^ _ ^ ^, ^ ^ ^^^ - _ -g-_ ^ 

25 X 34 850 
Q 04 — "y2 ~ ^"^^ ~ ^^ ^^^^ ^^ inches nearly. 

In irregular bodies with plain sides, the centre of gravity 

22 



170 



A3>IERICAN HOUSE-CARPENTER. 



may be found by balancing them upon tbe edge of a prism — 
upon the edge of a table — in two positions, making a line each 
time upon tlie body in a line with the edge of the prism, and 
the intersection of those lines will indicate the point required. 
Or suspend the article by a cord or thread attached to one 
corner or edge ; also, from the same point of suspension, hang 
a plumb-Une, and mark its position on the face of the article ; 
again, suspend the article from another corner or side, (nearly 
at right angles to its former position,) and mark the position 
of the plumb-line upon its face ; then the intersection of the 
two lines will be the centre of gravity. 




Fig. 216. 



277. — The effect of the weight of inclined learns. An in- 
clined post or strut, supporting some heavy pressure applied at 
its upper end, as at Fig. 209, exerts a pressure at its foot in 
the direction of its length, or nearly so. But when such a 
beam is loaded uniformly over its whole length, as the rafter 
of a roof, the pressure at its foot varies considerably from the 
direction of its length. For example, let A B, {Fig. 216,) be 
a beam leaning against the wall, B c, and supported at its 
foot by the abutment, A^ in the beam, A c, and let o be the 
centre of gravity of the beam. Through <?, draw the vertical 
line, 5 6?, and from B^ draw the horizontal line, B h, cutting 
h dinh ; join h and A, and h A will be the direction of the 
thrust. To prevent the beam from loosing its footing, the joint 
at A should be made at right angles to t A. The amount of 
pressure will be found thus : let h d^ (by any scale of equai 



FRAMING. 



171 



jaits,) equal tlie number of tons upon the beam, A B ; dra^v 
d e^ parallel to B I ; then I e, (by the same scale,) equals the 
pressure in the direction, l A ; and e d^ the pressure against 
the wall at B — and also the horizontal thrust at A^ as these 
are always equal in a construction of this kind. 

278. — The horizontal thrust of an inclined beam, {Fig. 216,) 
— the effect of its own weight — may be calculated thus ; 

Eule. — Multiply the weight of the beam in pounds by its 
base, A C^ in feet, and by the distance in feet of its centre of 
gravity, <9, (see Art, 275 and 276,) from the lower end, at A ; 
and divide this product by the product of the length, A B^ 
into the height, B C, and the quotient will be the horizontal 

fj T) 7/? 

thrust in pounds. This may be stated thus : H = —j-y-i where 

d equals the distance of the centre of gravity, (9, from the 

lower end ; h equals the base, A G ; w equals the weight of 

the beam ; h equals the height, B G ; I equals the length of 

the beam ; and ^equals the horizontal thrust. 

Example. — A beam, 20 feet long, weighs 300 pounds; its 

centre of gravity is at 9 feet from its lower end ; it is so 

inclined that its base is 16 feet and its height 12 feet ; wdiat is 

the horizontal thrust ? 

o- dlio , 9 X 16 X 300 9 X 4 X 25 „ , . 

Here —^t- becomes = =: 9 x 4 x o 

hi 12 X 20 5 

= 180 — H — the horizontal thrust. 

This rule is for cases where the centre of gravity does not 
occur at the middle of the length of the beam, although it is 
applicable when it does occur at the middle ; yet a shorter 
rule will suffice in this case, — and it is thus : — ■ 

Bide. — Multiply the weight of the rafter in pounds by the 
base, A G, {Fig. 216,) in feet, and divide the product by twice 
the height, B (7, in feet ; and the quotient will be the horizon 
tal thrust, when the cer tre of gravity occurs at the middle of 
the beam. 



172 



AMKRICAN HOUSE-OARPENTER. 



If the inclined beam is loaded, with an equally distributed 
load, add this load to the weight of the beam, and use this 
total weight in tht rule instead of the weight of the beam. 
And generally, if the centre of gravity of the combined 
weights of the beam and load does not occur at the centre of 
the leno:th of the beam then the former rule is to be used. 




Fig. 217. 



279. — In Fig. 217, two equal beams are supported at their 
feet by the abutments in the tie-beam. This case is similar to 
the last ; for it is obvious that each beam is in precisely the 
position of the beam in Fig. 216. The horizontal pressures at 
^, being equal and opposite, balance one another ; and their 
horizontal thrusts at the tie-beam are also equah (See Art. 
2m— Fig. 209.) When the height of a roof, {Fig. 217,) is 
one-fourth of the span, or of a shed, {Fig. 216,) is one-half the 
span, the horizontal thrust of a rafter, whose centre of gravity 
is at the middle of its length, is exacdy equal to the weight 
distributed uniformly over its surface. 




\^r^/-JW 



fig 218. 



FKAMING. iTeS 

280. — In shed, or lean-to roofs, as Fig. 216, l.ae horizontal 
pressure will be entirely removed, if the bearings of the raft- 
ers, as A B^ {Fig. 218,) are made horizontal — provided, how- 
ever, that the rafters and other framing do not bend between 
the points of support. If a beam or rafter have a natural 
curve, the convex or rounding edge shonld be laid uppermost. 

281. — A beam laid horizontally, supported at each end and 
nniformly loaded, is subject to the greatest strain at the mid- 
dle of its length. Hence mortices, large knots and other de- 
fects, shonld be kept as far as possible from that point ; and, 
in resting a load upon a beam, as a partition upon a floor 
beam, the weight should be so adjusted, if possible, that it will 
bear at or near the ends. 

Twice the weight that will break a beam, acting at the 
centre of its length, is required to break it when equally dis- 
tributed over its length ; and precisely the same deflection oi 
sag will be produced on a beam by a load equally distributed, 
that five-eighths of the load w411 produce if acting at the centre 
of its length. 

282. — "When a beam, supported at each end on horizontal 
bearings, (the beam itself being either horizontal or inclined,) 
has its load equally distributed, the amount of pressure caused 
by the load on each point of support is equal to one half the 
load ; and this is also the case, when the load is concentrated 
at the middle of the beam, or has its centre of gravity at the 
middle of the beam ; but, when the load is unequally distri- 
buted or concentrated, so that its centre of gravity occurs at 
some other point than the middle of the beam, then the amount 
of pressure caused by the load on one of the points of support 
is unequal to that on the other. The precise amount on each 
may be ascertained by the following rule. 

Bute. — Multiply the weight w^ {Fig. 219,) by its distance, CB, 
from its nearest point of support, B, and divide the product 
by the length, A B^ of the beam, and the quotient wil be the 



174 



AMERICAN HOTJSE-CARPENTER. 



-^\^ 

k 



Fig. 219. 

amount of pressure on the remote point of support, A. Again, 
deduct this amount from the weight, Ws and the remainder 
will be the amount of pressure on the near point of support, 
B ; or, multiply the weight, w^ by its distance, A 0^ from the 
remote point of support, JL, and divide the product by the 
length, A B^ and the quotient will be the amount of pressure 
on the near point of support, B. 

When I equals the length, A B ', a = A C ; h — G B^ and 
w = the load, then 

~j~— A = the amount of pressure at A, and 

-J- = B = the amount of pressure at B. 

Examjple. — A beam. 20 feet long between tlie bearings, has 
a load of 100 pounds concentrated at 3 feet from one of the 
bearings, what is the portion of this weight sustained by each 
bearing ? 

Here 'wj = 100 ; a,Vl\ l,Z\ and Z, 20. 



Hence J- =■ 



w 



l 100 X 3 



I - 



20 



= 15. 



, ^ w a 100 X 17 

And^=— -=— 2y-=85. 

Load on ^ = 15 pounds. 
Load on ^ = 85 pounds. 
Total weight = 1^ pounds. 



TEAMING. 175 

RESISTANCE OF MATERIALS. 

283. — Before a roof truss, or other piece c f framing, can be 
properly designed, two things are required to be known. The 
one is, the effect of gravity acting upon the yarions parts of 
the intended structure ; the other, the power of resistance 
possessed by the materials of which the framing is to be con- 
structed. In the preceding pages, the former subject having 
been treated of, it remains now to call attention to the latter. 

284. — Materials used in construction are constituted in their 
structure either of fibres (threads) or of grains, and are termed, 
the former fibrous, the latter granular. Ml woods and wrought 
metals are fibrous, while cast iron, stone, glass, &c., are gra- 
nular. The strength of a granular material lies in the power 
of attraction, acting among the grains of matter of which the 
material is composed, by which it resists any attempt to sepa- 
rate its grains or particles of matter. A fibre of wood or of 
wrought metal has a strength by which it resists being com- 
pressed or shortened, and finally crushed ; also a strength by 
which it resists being extended or made longer, and finally 
sundered. There is another kind of strength in a fibrous mate- 
rial ; it is the adhesion of one fibre to another along their sides, 
or the lateral adhesion of the fibres. 

285. — In the strain applied to a piece of timber, as a post 
supporting a weight imposed upon it, [Fig. 220,) we have an 
instance of an attempt to shorten the fibres of which the tim- 
ber is composed. The strength of the timber in this case is 
termed the resistance to compression. In the strain on a piece 
of timber like a king-post or suspending piece, (JL, Fig. 221,) 
we have an instance of an attempt to extend or lengthen the 
fibres of the material. The strength here exhibited is termed 
the resistance to tension. When a piece of timber is strained 
like a floor beam, or any horizontal piece carrying a load, 
(Fig. 222,) we have an instance in which the two strains of 



176 



AMEEIOAN HOTJSE-CARPENTEE. 






compression and tension are brought into action ; the fibres of 
the upper portion of the beam being compressed, and those of 
the under part being stretched. This kind of strength of tim 
ber is termed resistance to cross strains. In each of these three 
kinds of strain to which timber is subjected, the power of 
resistance is in a measure due to the lateral adhesion of the 
fibres, not so much perhaps in the simple tensile strain, yet to 
a considerable degree in the compressive and cross strains. 
But the power of timber, by which it resists a pressure acting 
compressively in the direction of the length of the fibres, tend- 
ing to separate the timber by splitting off a part, as in the 
case of the end of a tie beam, against which the foot of 'the 
rafter presses — is wholly due to the lateral adhesion of the 
fibres. 

286. — ^The strength of materials is that power by which they 
resist fracture^ while the stiffness of materials is that quality 
which enables them to resist deflection or sagging. A know- 
ledge of their strength is useful, in order to determine their 



FEAMING. 177 

iiinits of size to sustain given weights safely ; but a knowledge 
of their stiffness is more important, as in almost all construe 
tions it is desirable not only that the load be safely sustained, 
but that no appearance of weakness be manifested by any sen- 
sible deflection or sagging. 

I. RESISTANCE TO COMPRESSION. 

287. — The resistance of materials to the force of compression 
may be considered in four several ways, viz. : 

1st. When the pressure is applied to the fibres longitudi 
nally, and on short pieces. 

2d. When the pressure is applied to the fibres longitudi- 
nally, and on long pieces. 

3d. When the pressure is applied to the fibres longitudi- 
nally, and so as to split off the part pressed against, causing 
the fibres to separate by sliding. 

4th. When the pressure is applied to the fibres trans- 
versely. 

Posts having their height less than ten times their least side 
will crush before bending ; these belong to the first case : 
while posts, whose height is ten times their least side, or hxitq 
than ten times, will bend before crushing ; these belong to the 
second case. 

288. — In the above first and fourth cases of compression, 
experiment has shown that the resistance is in proportion to 
the number of fibres pressed, that is, in proportion to the area. 
For example, if 5,000 pounds is required to crush a prism with 
a base 1 inch square, it will require 20,000 pounds to crush a 
prism having a base of 2 by 2 inches, equal to 4 inches area ; 
because 4 times 5,000 equals 20,000. Experiment has also 
shown that, in the third case, the resistance is in proportion to 
the area of the surface separated without regard to the form 
of the surface. 

289. — In the second case of compression, the resistance is in 

23 



178 AMEKICAN HOUSE-CAEPENTEE, 

proportion to the area of the cross section of the piece, miiiti 
plied by the square of its thickness, and inversely in propor- 
tion to the square of the length, multiplied by the weight. 
When the piece is square, it will bend and break in the direc- 
tion of its diagonal ; here, the resistance is in proportion to the 
square of the diagonal multiplied by the square of the dia- 
gonal, and inversely proportional to the square of the length 
multiplied by the weight. If the piece is round or cylindrical, 
its resistance will be in accordance with the square of the dia- 
meter multiplied by the square of the diameter, and inversely 
proportional to the square of the length, multiplied by the 
weight. 

290. — ^These relations between the dimensions of the piece 
strained and its resistance, have resulted from the discussion 
of the subject by various authors, and rules based upon these 
relations are in general use, yet their accuracy is not fully 
established. Some experiments, especially those by Prof 
Hodgkinson, have shown that the resistance is in proportion to 
a less power of the diameter, and inversely to a less power of 
the height ; yet the variance is not great, and inasmuch as the 
material is restricted in the rules to a strain decidedly within 
its limits of resistance, no serious error can be made in the 
use of rules based on the aforesaid relations. 

291. — Experiments. In the investigation of the laws appli- 
cable to the resistance of materials, only such of the relations 
of the parts have been considered as apply alike to wood and 
metal, stone and glass, or other material, leaving to experi- 
ment the task of ascertaining the compactness and cohesion of 
particles, and the tenacity and adhesion of fibres ; those quaU- 
ties upon which depend the superiority of one kind of material 
over another, and which is represented in the rules by a constant 
number, each specific kind of material having its own special 
constant^ obtained by experimenting on specimens of that 
peculiar material. 



FRAMING. 



179 



292. — The following table exhibits the results of experiments 
on sucli woods as are in most common use in this country for 
the purpose of construction. The resistance of timber of the 



TABLE I. COMPKESSION. 







a 




^k 




i, 

a 


<p 






o 




o p. 




cs^ 


•O <0 








<D 


r— <0 


® 


'-'S 


■^ F^ 




i 


S 


^ 


■2S 


5 


CO a 




Kind of Material. 


1 

6 

o 

•3 


11 


ii 


m 
III 




|1 




lis 




(» 


H 


^ 


PM 


> 


H 


k 






Pounds 




Pounds 




Pounds 








per in. 




per in. 




per in. 




White wood, 


•397 


•2432 


600 






600 


3!)0 


Mahogan}' (Baywood), 


•439 


3527 


880 






1300 


650 


Ash, ..'... 


•517 


4175 


1040 






2300 


1150 


Spruce, 


. 


•369 


4199 


1050 


470 


160 


500 


250 


Chestnut, . 


. 


•491 


4791 


1200 


690 


230 


950 


475 


White pine, 


. 


•388 


,4806 


120O 


490 


160 


600 


300 


Ohio pine. 


, 


•586 


4809 


1200 


388 


130 


1250 


625' 


Oak, . 


, 


•612 


5316 


1330 


780 


260 


1900 


950 


Hemlock, . 


. 


•423 


5400 


1350 


540 


180 


600 


300 


Black walnut, 


. 


•421 


6594 


1400 






1600 


800 


Maple, 


. 


•574 


6061 


1515 






2050 


1025 


Cherry, . ■ 


. 


•494 


6477 


1620 






1900 


950 


Whiteoak, 


, 


•774 


6660 


1665 






2000 


1000 


Georgia pine, 


, 


•613 


6767 


1700 


510 


170 


1700 


850 


Locust, 


, 


•762 


7652 


1910 


1180 


400 


2100 


1050 


Live oak, . 


. 


•916 


7936 


1980 






5100 


2550 


Mahogany (St. ] 


Domingo), . 


•837 


8280 


2070 






4300 


2150 


Lignum vitae, 




1-282 


8650 


2160 






5800 


2900 


Hickory, . 




•877 


9817 


2450 






3100 


1550 



same name varies much ; depending as it obviously must on 
the soil in which it grew, on its age before and after cutting, 
on the time of year when cut, and on the manner in which it 
has been kept since it was cut. And of wood from the same 
tree, much depends upon its location, whether at the butt or 
towards tlie limbs, and whether at the heart or at the sap, or 
at a point midway from the centre to the circumference of the 
tree. The pieces submitted to experiment were of ordinary 
good quality, such as would be deemed proper to be used in 
framing. The prisms crushed were 2 inches long, and from 1 
inch to 1 J inches square ; some were wider one way than the 



180 AMEEICAN HOUSE-CAEPENTEK. 

other, but all containing in area of cross section from 1 to 2 
inches. There were generally three specimens of eacl* kind. 
The weio^ht g^iven in the table is the averas^e crushing: weio-ht 
per superficial incli. 

In the preceding table the first column contains the specific 
gravity of the several kinds of \vood, showing their compara- 
tive density. The weight in pounds of a cubic foot of any 
kind of wood or other material, is equal to its specific gravity 
multiplied by 62*5 ; this number being the weight in pounds 
of a cubic foot of water. The second cohimn contains the 
weight in pounds required to crush a prism having a base of 
one inch square ; the pressure applied to the fibres longitudi- 
nally. The third column contains the value of G in the rules ; 
C being equal to one-fourth of the crushing weight in the 
preceding column. The fourth column contains the w^eight 
in pounds, which, applied to the fibres longitudinally, is 
required to force off a part of the piece, causing the fibres to 
separate by sliding, the surface separated being one inch 
square. The fifth column contains the value of IT in the 
rules, H being equal to one third of the weight in the preced- 
ing column. The sixth column contains the weight in pounds 
required to crush the piece when the pressure is applied to the 
fibres transversely, the piece being one inch thick, and the 
surface crushed being one inch square, and depressed one 
twentieth of an inch deep. The seventh column contains the 
value of Pin the rules; 7-* being the weight in pounds applied 
to the fibres transversely, which is required to make a sensible 
impression one inch square on the side of the piece, this being 
the greatest weight that would be proper for a post to be 
loaded with per inch surface of bearing, resting on the side of 
the kind of wood set opposite in the table. A greater weight 
would, in proportion to the excess, crush the side of the wood 
under the post, and proportionably derange the framing, if not 
cause a total failure. It will be observed that the measure oi 



FRAMING. 18 J 

tnis resistance is useful in limiting the load on a post accord- 
ing to the kind of material contained, not in the jpost^ but in 
the timber upon which the post presses. 

293. — Tn Table II. are the results of experiments made to 
test the resistance of materials to flexure : first, the flexuro 
produced by compression, the force acting on the ends of the 
flbres longitudinally ; secondly, the flexure arising from the 
efl'ects of a cross strain, the force acting on the side of the 
fibves transversely, the beams Iseing laid on chairs or rests. 
Of white oak, No. 1, there were eight specimens, of 2 by 4 
Indies, and 3|- feet long, seasoned more than a year after they 
were prepared for experiment. Of the other kinds of wood 
there w^ere from three to five specimens of each, of 1:? by 2:? 
inches, and from W to 2f feet long. Of the cast iron there 
were six specimens, of L inch square and 1 foot long; and 
of the wrought iron there were five specimens of American, 
three of f by 2 inches, and two of l-J inches square, and three 
specimens of common English, ^ by 2 inches ; the eight speci- 
mens being each 19 inches long, clear bearing. In each 
case the result is the average of the stifi'ness of the several 
specimens. The numbers contained in the second column are 
the weights producing the first degree of flexure in a post or 
strut, where the post or strut is one foot long and one inch 
square; so, likewise, the numbers in the fifth column, and 
which are represented in the rules by E^ are the weights 
required to deflect a beam one inch, where the beam is one 
foot long, clear bearing, and one inch square. — (See remarks 
upon this, Art. (321.) The numbers in the third column are 
equal to one-half of those in the second. The numbers con- 
tained in the fourth column, and represented by n in the 
rules, show the greatest rate of deflection that the material 
may be subjected to without injury. This rate multiplied by 
the length in feet, equals the total deflection within the limits 
of elasticity. 



182 



AMERICAN HOTJSE-CAEPENTER. 



TABLE II. FLEXURE. 





Specific 
Gravity. 


Under 
Compression. 


Under 
Cros6 Strain. 


Kind of Material. 


Pounds pro- 
ducing the 
first degree 
of flexure. 


Value of 

^in 
the Eules. 


Value of 

n in 
the Eules. 


Value >t 

Eir 
the Eules 


Hemlock, .... 


0-402 


2640 


1320 


08794 


1240 


Spruce, 


•432 


4190 


2095 


0-09197 


1550 


White pine, .... 


•407 


2350 


1175 


0-1022 


1750 


Ohio yellow pine, . 


•586 


6000 


3000 


0-049 


1970 


Chestnut, 


•52 


7720 


3860 


0-07541 


2330 


White oak, No. i, . 


•82 






009152 


2520 


White oak, No. 2, . 


•805 


6950 


3475 


0-0567 


2590 


Georgia pine, .... 


•755 


9660 


4830 


0-07723 


2970 


Locust, 


•863 


10920 


5460 


0-066 15 


3280 


Cast iron, .... 


7-042 






00148 


30500 


Wrought iron, common English, 


7-576 






0-03717 


45500 


Wrought iron, American, . 


7-576 






0-04038 


51400 



PRACTICAL RULES FOR COMPRESSION". 



First Case. 



294. — To find the weight that can be safely sustained by a 
post, when the height of the post is less than ten times the 
diameter if round, or ten times the thickness if rectangular, 
and the direction of the pressure coinciding with the axis. 

Rule I. — Multiply the area of the cross-section of the post, 
in inches, by the value of G in Table I., the product will be 
the required weight in pounds. 

AG=w. ■ (1.) 

Example. — A Georgia pine post is 6 feet high, and in cross- 
section, 8 X 12 inches, what weight will it safely sustain ? 
The area = 8 x 12 = 96 inches ; this multiplied by lYOO, the 
value of (7, in the table, set opposite Georgia pine, the result, 
163,200, is the weight-in pounds required. It will be observed 
that the weight would be the same for a Georgia pine post of 
any height less than 10 times 8 inches — 80 inches == 6 feet 8 



FKAMINa. 1 83 

inches, provided its breadth and thickness remain the same, 
12 and 8 inches. 

295.— To find the area of the cross-section of a post to sus- 
tain a given weight safely, the height of the post being less 
than ten times the diameter if round, or ten times the least 
side if rectangular ; the pressure coinciding with the axis. 

Hide II. — Divide the given weight in pounds by the value of 
6', in Table I., and the product will be the required area in inches 

Example, — A weight of 38,400 pounds is to be sustained by 
a white pine post 4 feet high, what must be its area of section 
in order to sustain the weight safely ? Here, 38,400 divided 
by 1200, the value of G^ in Table I., set opposite white pine, 
gives a quotient of 32 ; this, therefore, is the required area, 
and such a post may be 5 X 6'4 inches. To find the least side, 
so that it shall not be less than one-tenth of the height, divide 
the height, reduced to inches, by 10, and make the least side 
to exceed this quotient. The area, divided by the least side 
so determined, will give the wide side. If, however, by this 
process, the first side found should prove to be the greatest, 
then the size of the post is to be found by Rule YIL, YIIL, or 
IX. 

296. — If the post is to be round, by reference to the Table 
of Circles in the Appendix, the diameter will be found in the 
column of diameters, set opposite to the area of the post found 
in the column of areas, or opposite to the next nearest area. 
For example, suppose the required area, as just found by the 
example under Rale II., is 32 ; by reference to the column of 
areas, 33'183 is the nearest to 32, and the diameter set opposite 
is 6-5. The post may, therefore, be Q^ inches diameter. 

Second Case, 
29 Y.- -To ascertain the weiglit that can be sustained safely 



184 AMERICAN HOUSE-CAEPENTEE. 

by a post whose height is, at least, ten times its least side if 
rectangular, or ten times its diameter if round, the directiou 
of the pressure coinciding with the axis. 

JRule III. — ^Yherh the jpost is round the weight may he 
found by this rule : Multiply the square of the diameter in 
inches by the square of the diameter in inches, and multiply 
the product by 0*589 times the value of B^ in Table IL, divide 
this product by the square of the height in feet, and the quo- 
tient will be the required weight in pounds. 

0-5S9 BB'I)' 0-6S9BI)' 
w= ^, =. J, (3.) 

Example. — What weight will a Georgia pine post sustain 
safely, whose diameter is 10 inches and height 10 feet ? The 
square of the diameter is 100 ; 100 x 100 =. 10,000. And 
10,000 by 0-589 times 4830, the value of B, Table IL, set 
opposite Georgia pine, =: 28,448,700, and this divided by 100, 
the square of the height, equals 284,487, the weight required, 
in pounds. 

Mule TV. — If the post he rectangular the weight is found 
by this rule : Multiply the area of the cross-section of the post 
by the square of the thickness, both in inches, and by the 
value of B^ Table IL Divide the product by the square of 
the height in feet, and the quotient will be the required 
weight in pounds. 

AfB I t^ B 

Example. — What weight will a white pine post sustain 
safely, whose height is 12 feet, and sides 8 and 12 inches re- 
spectively ? The area = 8 x 12 = 96 inches ; the square of 
the thickness, 8, = 64. The area by the square of the thick- 
ness, 96 X 64, ^ 6144 ; and this by 1175, the value of B^ for 
white pine, equals 7,219,200. This, divided by 144, the 
square of the he'ght = 50,133^, the required weight in 
pounds. 






FRAMmG. 



tfe-?^*^ 



p^ule Y.—If the post le square, the weight is found by this / ^n i^ 
rule : Multiply the value of ^, Table IL, by the square of the (^^-<^". 

area of the post in inches, and divide the product by the ^J -r 

square of the height in feet, and the quotient will be the .i^^V*^ '^ 
required weight in pounds. i^jJL--^ 

w = —r— = -T— . (5.) ^^^ ut<i ^^- 



h' - h' • ^"-^ " H/ 

Exam/ple. — What weight will a white oak post sustain 
safely, whose height is 9 feet, and sides each 6 inches ? The 
value of B, set opposite white oak, is 3475 ; this, by (36 x 36 
=) 1296, tlie square of the area, equals 4,503,600. This pro- 
duct, divided by 81, the square of the heiglit, gives for quo- 
tient, 55,600, the required weight in pounds. 

298. — ^To ascertain the size of a post to sustain safely a given 
weight when the height of the post is at least ten times the 
least side or diameter. 

Bule YI. — When the post is to he round or cylindrical, the 
size may be obtained by this rule : Divide the weight in 
pounds by 0*589 times the value of B, Table IL, and extract 
the square root of the product ; multiply the square root by 
the height in feet, and the square root of this product will be 
the diameter of the post in inches. 




^ V-5895 V- 



589 S 



(0.) 



Example. — What must be the diameter of a locust post, 10 
feet high, to sustain safely 40,000 pounds? Here 0-589 times 
5,460, the value of B for locust. Table IL, equals 3215*9. 
The weight, 40,000, divided by 3215*9, equals 12*438. The 
square root of this, 3*5268, multiplied by 10, the height, equals 
35*268, and the square root of this is 5*9386 or 5|f inches, the 
required diameter of the post. 

Bule YII. — If the post is to he rectangular, the size may be 
obtained by this rule : Multiply the square of the height in 

24 



186 AMERICAN HOUSE-CAKPENTER. 

feet by tlie weight in pounds, and divide the product by the 
value of B^ Table II. Now, if the breadth is known, divide 
the quotient by the breadth in inches, and the cube root of 
this quotient will be the thickness in inches. But if the thick- 
ness is known, and the breadtli desired, divide, instead, by the 
cube of the thickness in inches, and the quotient will be the 
breadth in inches. 



'"Vs-j (70 

» = ^ w 

Exam,])le. — What thickness must a hemlock post have, 
whose breadth is 4 inches and height 12 feet, to sustain safely 
1,000 pounds ? The square of the height equals IM ; this, by 
1 ^000, the weight, equals 144,000. This, divided by 1,320, the 
value of B for hemlock. Table IL, equals 109*091. This, 
divided by 4, the breadth, equals 27*273, and the cube root 
of this is 3'01, a trifle over 3 inches, and this is the thickness 
required. 

Another Example. — What breadth must a spruce post have, 
whose thickness is 4 inches and height 10 feet, to sustain safely 
10,000 pounds? The square of the height, 100, by 10,000, the 
weight, equals 1,000,000. This, divided by 2095, the value of 
B. Table II., for spruce, equals 477-09 ; and this, divided by 
64, the cube of the thickness, equals 7'45, nearly 7-i- inches, 
the breadth required. 

Bule Yin. — If the post is to he square, the size may be 
obtained by this rule. Divide the weight in pounds by the 
value of ^, Table II., and multiply the square root of the pro- 
duct by the height in feet, and the square root of this product 
will be the dimension of a side of the post in inches. 

Examjple. — What dimension must the side of a square post 



FEAMING. 187 

have, whose lieiglit is 15 feet, the post being of Georgia 
pine, to sustain safely 50,000 pounds? The weight 50,000, 
divided by 4830, the value of B^ Table XL, for Georgia pine, 
equals 10-352. The square root of this, 3*2175, multiplied by 
15, the height, equals 48-362, and the square root of this is 
6-9472, nearly 7 inches, the size of a side of the required 
post. 

299. — A square post is not the stiffest that can be made from 
a given amount of material. The stiffest rectangular post is 
that whose sides are in proportion as 6 is to 10. When this 
proportion is desired it may be obtained by the following rule. 

B,%ble IX. — Divide six-tenths of the weight in pounds by the 
value of B^ Table II., and extract the square root of the quo- 
tient ; multiply the square root by the height in feet, and then 
the square root of this product will be the thickness in inches. 
The breadth is equal to the thickness divided by 0-6. 

/ - \ K / £^ (/o-6 h\ w (10.) 

^~ ^ B ~^ B 

5^.— (11-) 

0-6 ^ ^ 

Exaw,])le. — What must be the breadth and thickness of a 
white pine post, 10 feet high, to sustain safely 25,000 pounds. 
Here ^^ of 25,000, the weight, divided by 1175, the value of 
B^ Table IL, for white pine, equals 12-766. The square root 
of this, 3-5729, multiplied by 10, th i l;eight, equals 35*729, and 
the square root of this is 5*977, nearly 6 inches, the thickness 
required. Th s, divided by 0-6, equals 10, equals the breadth 
in inches required. 

300. — The sides of a post may be obtained in any desirable 
proportion by Kule IX., simply by changing the decimal 0*6 
to such decimal as will be in proportion to unity as one side is 
to be to the other. For example, if it be desired to have the 
Bides in proportion as 10 is to 9, then 0*9 is the required 
decimal; if as 10 is to 8, then 0-8 is the decimal; if as 10 



188 AMEKICAN HOUSE-CAKPENTEE. 

is to Y, then 0*7 is the decimal to be used in pl/ace of 0*6 
in the rule. And generally let h equal the broad side and i 
the narrow side, or let these letters represent respectively the 
numbers that the sides are to be in proportion to ; then, where 

X equals the decimal sought, h \ t wXx x~ — ^ the required 

h 

decimal, ov fraction. For a fraction may be used in place of 
the decimal, where it would be more convenient, as is the case 
when the sides are desired to be in proportion as 3 to 2. Here 
3 : 2 :: 1 : a? = §. This fraction should be used in the rule in 
place of the decimal 0*6 — rather than its equivalent decimal; 
simply because the decimal contains many figures, and there- 
fore would not be convenient. The decimal equivalent to f is 

Third Case, 

301. — To ascertain what weight may be sustained safely 
by the resistance of a given area of surface, when the weight 
tends* to split off the part pressed against by causing one sur- 
face to slide on the other, in case of fracture. 

Rule X. — Multiply the area of the surface by the value of 
/7, in Table L, and the product will be the weight required in 
pounds. 

AH=w. (12.) 

Example. — The foot of a rafter is framed into the end of 
its tie-beam, so that the uncut substance of the tie-beam is 15 
inches long from the end of the tie-beam to the joint of the 
rafter ; the tie-beam is of white pine, and is six inches thick ; 
what amount of horizontal thrust will this end of the tie-beam 
sustain, without danger of having the end of the tie-beam split 
off? Here the area of surface that sustains the pressure is 6 
by 15 inches, equal to 90 inches. This, multiplied by 160, the 
value of H^ set opposite to white pine Table I., gives a product 
of 14,400, and this is the required weight m pounds. 



TEAMING. 189 

302. — To ascertain tlie area of surface that is required to 
sustain a given weight safely, when the weight tends to split 
off the part pressed against, by causing one surface to slide on 
the other, in case of fracture. 

B,ule XI. — Divide the given weight in pounds by the value 
of H^ Table I., and the quotient will be the required area in 
inches. 

^ = 5. (18.) 

Example. — The load on a rafter causes a horizontal thrust at 
its foot of 40,000 pounds, tending to split off the end of the tie- 
beam, what must be the length of the tie-beam beyond the 
line, where the foot of the rafter is framed into it, the tie-beam 
being of Georgia pine, and nine inches thick ? The weight, or 
horizontal thrust, 40,000, divided by 170, the value of H^ 
Table L, set opposite Georgia pine, gives a quotient of 235*3. 
This, the area of surface in inches, divided by 9, the breadth 
of the surface strained, (equal to the thickness of the tie-beam,) 
the quotient, 26'1, is the length in inches from the end of 
the tie-beam to the rafter joint, say 26 inches. 

303. — A knowledge of this kind of resistance of materials is 
useful, also, in ascertaining the length of framed tenons, so as 
to prevent the pin, or key, with which they are fastened from 
tearing out ; and, also, in cases where tie-beams, or other 
timber under a tensile strain, are spliced, this rule gives the 
\ength of the joggle on each end of tlie splice. 

Fourth Case. 

304. — ^To ascertain what weight a post may be loaded with, 
so as not to crush the surface against wliich it presses. 

Rule XII. — Multiply the area of the post in inches by the 
value of P, Table L, and the product is the weight required in 
pounds, 

w^A P. (14.) 



190 AMERICAN H0U8E-CAKPENTEE. 

Example. — A post, 8 by 10 inches, stands upon a white pno 
girder; the area eqnals 8 x 10 = 80 inches. This, by 300, tlie 
value of P, Table L, set opposite white pine, the product, 
24,000, is the required weight in po mds. 

305. — To ascertain what area a post must have in order to 
prevent the post, loaded wdtli a given weight, from crushing 
the surface against which it presses. 

Hide XIII. — Divide the given weight in pounds by the value 
of P, Table I., and the quotient will be the area required in 
inches. 

A=^,. (15.) 

Emm^ple. — A post standing on a Georgia pine girder is 
loaded with 100,000 pounds, what must be its area? The 
weight, 100,000, divided by 850, the value of P, Table I., set 
opposite Georgia pine, the quotient, 117*65, is the required 
area in inches. The post may be 10 by llf , or 10 x 12 inches, 
or, if square, each side will be 10'84: inches, or 12^ inches 
diameter, if round. 

II. RESISTANCE TO TENSION. 

306. — The resistance of materials to the force of stretching, 
as exemplified in the case of a rope from which a weight is 
suspended, is termed the resistance to tension. In fibrous 
materials, this force will be dififerent in the same specimen, in 
accordance with the direction in which the force acts, whether 
in the direction of the length of the fibres, or at right angles to 
the direction of their length. It has been found that, in hard 
woods, the resistance in the former direction is about 8 to 10 
times what it is in the latter; and in soft woods, straight 
grained, such as white pine, the resistance is from 16 to 20 
times. A knowledge of the resistance in the direction of the 
fibres is the most useful in practice. 

307.- -In the following table, Xho, experiments recorded were 



FRAMING. 



19-i 



to test tliis resistance in sncli woods, also iron, as are in common 
use. Each specimen was turned cylindrical, and about 2 
inches diameter, and then the middle part for 10 inches in 
length reduced to f ths of an inch diameter, at the middle of 
the reduced part, and gradually increased toward each end, 
where it was about an eighth of an inch larger at its junction 
with the enlarged end. 



TABLE ni. TENSION". 





Specific 
Gravity. 


Weight produc- 


Value of 


Kind of Material. 


ing fracture per 


T 




square inch. 


in the Eules. 






Pounds. 




Hickory, . . . 


0-751 


20,700 


3,450 


Locust, 


•794 


15,900 


2,650 


Maple, 


'694 


16,400 


2,567 


White pine, 


•458 


14,200 


2,867 


Ash, 


•608 


11,700 


1,950 


Oak, 


•728 


10,000 


1,667 


White oak, 


•774 


17,000 


2,888 


Georgia pine, 


•650 


17,000 


2,838 


castiron, If- ; ; ; : 


7-200 
7-600 


17,000 
30,000 


2,883 
5,000 


American wrought iron, 2 in. diam.. 


7-000 


80,000 


5,000 


Do. do. f and ^ do., 


7-800 


55,000 


9,166 


Do. do. wire, No. 3, . 




102,000 


17,000 


Do. do. do. No. 0, . 




74,500 


12,416 


Do. annealed do. No. 0, . 




53,000 


8,888 



308. — ^The value of T in the rules, as contained in the last 
column of the above table, is one-sixth of the w^eight pro- 
ducing fracture per square inch of cross section, as recorded in 
the preceding column. This proportion of the breaking 
weight is deemed the proper one, from the fact that in prac- 
tice, through defects in workmanship, the attachments Tnay be 
so made as to cause the strain to act along one side of the 
piece, instead of through its axis ; and as in this case it has 
been found, that fracture will be produced with § of the strain 
that can be sustained through the axis, therefore one half of 
this reduced strain, (equal to } of the strain through the axis), 
is tho largest that a due regard to security will permit to be 



192 AMERICAN HOUSE-CAEPENTER. 

used. And in some cases it may be deemed advisable to load 
the material with even a still smaller strain. 

309. — To ascertain the weight or pressure that mav be safely 
applied to a beam as a tensile strain. 

Bute XIY. — Multiply the area of the cross section of the 
beam in inches by the value of T^ Table III., and the product 
will be the required weight in pounds. The cross section here 
intended is that taken at the smallest part of the beam or rod. 
A beam is usually cut with mortices in framing ; the area will 
probably be smallest at the severest cutting : the area used in 
the rule must be only of the uncut fibres. 

AT=.w. (16.) 

Example. — A tie-beam of a roof truss is of white pine, and 
6 X 10 inches ; the cutting for the foot of the rafter reduces 
the uncut area to 40 inches : what amount of horizontal thrust 
from the foot of the rafter will this tie-beam safely sustain ? 
Here 40 times 2,367, the value of T, equals 94,680, the required 
weight in pounds. 

310. — To ascertain the sectional area of a beam or rod that 
will sustain a given weight safely, when applied as a tensile 
strain. 

Bule XY. — Divide the given weight in pounds by the value 
of T^ Table III., and the quotient will be the area required in 
inches : this will be the smallest area of uncut fibres. If the 
piece is to be cut for mortices, or for any other purpose, then 
a sufficient addition is to be made to the result found by 
the rule. 

- = X (17.) 

Examyle. — A rafter produces a thrust horizontally of 
80,000 pounds ; the tie-beam is to be of oak : what must the 
area of the cross section of the tie-beam be, in order to sustain 
the rafter safely? The given weight, 80,000, divided by 
1,667, the value of 1\ the quotient, 48, is the area of uncut 



FKAMING. 193 

fibres. This should have nsuall j one-half of its amount added 
to it as an allowance for cutting ; therefore 48 + 24 = Y2. 
The tie-beam may be 6 x 12 inches. 

311. — In these rules nothing has been said of an allowance 
for the weight of the beam itself, in cases where the beam is 
placed vertically, and the weight suspended from the end. 
Usually, in timber, this is small in comparison with the load, 
and may be neglected ; although in very long timbers, and where 
accuracy is decidedly essential, it may form a part of the rule. 

312. — Taking the effect of the weight of the beam into 
account, the relation existing between the weights and parts 
of the beam, may be stated algebraically thus : — • 
AT=w^-U (18.) 

Where A equals the area of the section of imcut fibres, T 
equals the tabular constant in the rules, which is equal to the 
load that may be safely trusted on a rod of like material with 
the beam and one inch square ; w equals the load, and 'k, 
equals the weight of the beam. Now, the w^eight of the beam 
equals its cubical contents in feet, multiplied by the weight of 
a cubic foot of like material ; and a cubic foot of the material 
equals 62*5 times its specific gravity, while the cubical contents 

of the beam in feet equals l^ where R equals the sectional 

area in inches, and I equals the leugth in feet. Hence — 

^ = 62-5/-j;^?, (19.) 

where y equals the specific gravity. It will be observed that 
A equals the sectional area of the uncut fibres, while B, equals 
the sectional area of the entire beam ; and, where the excess 
of R over A may be stated as a proportional part of A.^ or 
when A -\- n A — B,^ {n being a decimal in proportion ta 
unity, as the excess of R over A is to A,) or 

R — A 

—J— = n. Then, [from (18.) ] — 

25 



"5 94 AMERICAN HOUSE-CARPENTER. 

A T = w + k 



= . + 62-5^+;^/^ 




= ^-+^A{l + n)fl, 




= 'w + 0-i34{n + l)Afl; 




a.ndw = A T- 0-434 {n + l)Afl, 




10 = A (r- 0-434 {» + !)/;;) 


(20.) 


a°J ^ =7w-7T7ToT^^— TTVTT- 


(21.) 



When A is found, to find ^, we have from 

B = A +n A, 

J2 = A{n-hl.) (22.) 

As the excess of JR over A decreases, n also decreases, until 
finally, when R — A^n becomes zero. For — 

and when A = R^ then 

"When 7i equals zero, it disappears from the rules, and (20) 
becomes 

w = A (T- 0434/0 (23.) 

and (21) becomes 

^ = r_o'^434/Z, ^^*-) 

and (22) becomes 

R = A, (25.) 

313. — ^These rules stated in words at length are as fol- 
lows : — 

To ascertain the weight that may be suspended safely from 

a vertical beam, when the weight of the beam itself is to be 

taken into account, and when a portion of the fibres are cut in 

framing. 

RuU XYI. — ^From the sectional area of the beam, deduct 



FRAMING. 195 

the sectional area of uncut fibres, and divide the remainder by 
the sectional area of the uncut fibres, and to the quotient add 
unity; multiply this sum by 0434 times the specific gravity 
of the beam, and by its length in feet ; substract this product 
from tlie value of T^ Table III., and the remainder, multiplied 
by the sectional area of the uncut fibres, will be the required 
weight in pounds. 

w^A {T- 0-434 {n + 1)/ 1) (20.) 

Example. — A white pine beam, set vertically, 5x9 inches 
and 30 feet long, is so cut by mortices as to have remaining 
only 5x6 inches sectional area of uncut fibres : what w^eight 
will such a beam sustain safely, as a tensile strain ? The uncut 
fibres, 5x6 = 30, deducted from the area of the beam, 5x9 
= 45, there remains 15. This remainder, divided by 30, the 
area of the uncut fibres, the quotient is 0*5. This added to 
unity, the sum is 1'5. This, by 0*434 times 0*458, the specific 
gravity set opposite white pine in Table III., and by 30, the 
length of the beam in feet, the product is 8-95. This product, 
deducted from 2,367, the value of Tset opposite white pine in 
Table III., the remainder is 2,358*05. This remainder multi- 
plied by 30, the sectional area of the uncut fibres, the product, 
Y0,Y41'5, is the required weight in pounds. 

314. — When the beam is uncut for mortices or other pur- 
poses, the former part of {\\q rule is not needed ; the weight 
will then be found by the following rule. 

Rule XYII. — Deduct 0*434 times the specific gravity of 
the beam, multiplied by its length in feet, from the value 
of T^ Table III. ; the remainder, multiplied by the sectional 
area of the beam in inches, will be the required weight in 
pounds. 

w^A{T-0-4.Z4cfT). (23.) 

Ex(tm.ple. — A Georgia pine beam, set vertically, is 25 feet 
long and 7x9 inches in sectional area: what weight will 
it sustain safely, as a tensile strain ? By the rule, 0*434 times 



196 AJtlEKICAN HOUSE-CAKPENTEE. 

0*65, the specific gravity of Georgia pme, as in Table III., mul 
tiplied bj 25, the length in feet, the product is 7*05. Thia 
product, deducted from 2,833, the value of T^ Table III., set 
opposite Georgia pine, and the remainder, 2,825*95, multiplied 
by 63, the sectional area, the product, 178,034:'85, is the 
required weight in pounds. 

315. — To ascertain the sectional area of a vertical beam that 
will safely sustain a given tensile strain, where the weight of 
the beam itself is to be considered. 

Rule XYIII. — Where the beam is cut for mortices or other 
purposes, let the relative proportion of the uncut fibres to those 
that are cut, be as 1 is to n^ {n being a decimal to be fixed on 
at pleasure.) Then to the value of n add unity, and multiply 
ing the sum by 0'434: times the specific gravity in Table III., 
and by the length in feet. Deduct this product from tho 
value of T, Table III., divide the given w^eight in pounds by 
this remainder, and the quotient will be the area of the uncut 
fibres in inches. Add unity to the value of n^ as above, and 
multiply the sum by the area of the uncut fibres ; the product 
will be the required area of the beam in inches. 

■^ " r-0434(7?. + i)/^, ^^■^•^ 

E^A{n-V\\ (22.) 

Examjple. — A vertical beam of white oak, 30 feet long, is 
required to resist effectually a tensile strain of 80,000 pounds : 
what must be its sectional area ? The relative proportion of 
the uncut fibres is to be to those that are cut as 1 is to 0*4. 
To 0-4, the value oin, add 1* ; the sum is 1*4. This, by 0*434 
times '774, the specific gravity of white oak in Table III., and 
by 30, the length, the product is 14-109. This, deducted from 
2,833, the value of Tfor white oak in Table III., the remainder 
is 2,818'891. The given weight, 80,000, divided by 2,818-891, 
the remainder, as above, the quotient, 28*38, is the area of the 
uncut fibres. This multiplied by the sum of 0-4 and 1*, (or 



FRAMING. 197 

the value of 7i and unity =: 1'4,) the pre duct. 39'732, is the 
required area of the beam in inches. 

316. — When the fibres are uncut, then their sectional area 
equals the area of tie beam, and may be found by the follow- 
ing rule. 

Rule XIX, — Deduct 0434 times the specific gravity in 
Table III., multiplied by the length in feet, from the value of 
T, Table III., and divide the weight in pounds by the remain- 
der. The quotient will be the required area in inches. 

(24.) 



r-O-434/Z. 

Eiuimple. — A vertical beam of locust, 15 feet long, fibres all 
uncut, is required to sustain a tensile strain equal to 25,000 
pounds; what must be its area? Here 0-434 times '794, the 
specific gravity for locust in Table III., multiplied by 15, the 
length in feet, is 5*17. This, from 2,650, the value of T for 
locust, Table III., the remainder is 2,644'83. The given 
weight, 25,000, divided by 2,644*83, the remainder, as above, 
the quotient, 9"45, will be the required area in inches. 

III. RESISTANCE TO CROSS-STRAINS. 

317. — A load placed upon a beam, laid horizontal or in- 
clined, tends to bend it, and if the weight be proportionally 
large, to break it. The power in the material that resists this 
bending or breaking, is termed the resistance to Gross-strains^ 
or transverse strains. While in posts or struts the material is 
compressed or shortened, and in ties and suspending-pieces it 
is extended or lengthened; in beams subjected to cross-strains 
the material is both compressed and extended. (See Art 
254.) When the beam is bent, the fibres on the concave side 
are compressed, while those on the convex side are extended. 
The line where these two portions of the beam meet — that is, 
the portion compressed and the portion extended— the hori- 
zontal line of juncture, is termed the netttral line or plane. It 



198 AMERICAN HOUSE-CARPENTEK. 

is SO called because at this line the fibres are neither com 
pressed nor extended, and hence are under no strain whatever. 
The location of this line or plane is not far from the middle of 
the deptli of the beam, when the strain is not sufficient to 
injure the elasticity of the material ; but it removes towards 
the concave or convex side of tlie beam as the strain is 
increased, until, at the period .of rupture, its distance from the 
top of the beam is in proportion to its distance from the bot- 
tom of the beam as the tensile strength of the material is to its 
compressive strength. 

318. — In order that tlie strength of a beam be injured as 
little as possible by the cutting required in framing, all mor- 
tices should be located at or near the middle of the depth. 
There is a prevalent idea among some, who are aware that 
the upper fibres of a beam are compressed when subject to 
cross-strains, that it is not injurious to cut these top fibres, 
provided that the cutting be for the insertion of another piece 
of timber — as in the case of gaining the ends of beams into 
the side of a girder. They suppose that the piece filled in 
will as efi'ectually resist the compression as the part removed 
would have done, had it not been taken out. ^Now, besides 
the efi'ect of shrinkage, which of itself is quite sufficient to 
prevent the proper resistance to the strain, tliere is the mecha- 
nical difficulty of fitting the joints perfectly throughout ; and, 
also, a great loss in the power of resistance, as the material is 
so much less capable of resistance when pressed at right angles 
to the direction of the fibres, than when directly with them, as 
the results of the experiments in the tables show. 

319. — In treating upon tlie resistance to cross-strains, the 
subject is divided naturally into two parts, viz. stiffness and 
strength : tlie former being the power to resist deflection oi 
bending, and the latter the resistance to rupture. 

320. — Resistance to Deflection. When a load is placed 
upon a beam supported at each end, the beam bends more oi 



FRAMING. 



190 



less ; the distance that the beam descends iv.ider the operation 
of the load, measured at the middle of its length, is termed its 
deflection. In an mvestigation of the laws of deflection it has 
been demonstrated, and experiments have confirmed it, that 
while the elasticity of the material remains uninjured by the 
pressure, or is injured in but a small degree, the amount of 
deflection is directly in proportion to the weight producing it, 
and is as the cube of the length ; and, in pieces of rectangular 
sections, it is inversely proportional to the breadth and the 
cube of the depth : or, inversely proportional to the fourth 
power of the side of a square beam or of the diameter of a 
cylindrical one. Or, when I equals the length between the 
supports, w the weight or pressure, h the breadth, d the depth, 
and J) the deflection ; then — 

equals a constant quantity for beams of all dimensions made 
from a lihe material. Also, 

~=E, (27.) 

where s equals a side of a square beam ; and 

0^589~J9^ ^ ■^' ^'^^ 

where D equals the diameter of a cylindrical beam. The 
constant here is less than in the case of the square and of the 
rectangular beams. It is as much less as the circular beam is 
less stiff than a square beam whose side is equal to the diame- 
ter of the cylindrical one. The constant, E^ is therefore mul- 
tiplied by the decimal 0*589. 

321. — It may be observed that ^in (26) and (27) wonld be 
equal to %o^ in case the dimensions of the beam and the 
amount of deflection were each made equal to unity ; and in 
(28) equal to w divided by 0-589. That is, when in (26) the 
length is 1, the breadth 1, and the depth 1, then ^wou'd be 



200 AMERICAN HOUSE-CARPENTER. 

equal to the weight tliat would depress the beam from its ori 
ginal line equal to 1. Thus — 

V w _ V X w 

^ = hW^ - 1 X r X i = ^' 

the dimensions all taken in inches except the length, and this 
taken in feet. This is an extreme state of the case, for in most 
kinds of material this amount of depression would exceed the 
limits of elasticity ; and hence the rule would here fail to give 
the correct relation among the dimensions and pressure. For 
the law of deflection as above stated, (the deflection being 
equal for equal weights,) is true only while the depressions are 
small in comparison with the length. Nothing useful is, 
therefore, derived from this position of the question, except to 
give an idea of the nature of the quantity represented by the 
constant, E ', it being in reality a measure of the stiflfness of 
the kind of material used in comparing one material with 
another. Whatever maybe the dimensions of the beam, E^ 
calculated by (26,) will always be the same quantity for the 
same material; but when various materials are iised, ^ w^ill 
vary according to the flexibility or stifl'ness of each particular 
material. For example, ^will be much greater for iron thaa 
for wood ; and again, among the various kinds of wood, it w411 
be larger for the stiff woods than for those that are flexible. 

322. — If the amount of deflection that w^ould be proper in 
beams used in framing generally, (such as floor beams, girders 
and rafters,) w^ere agreed upon, the rules would be shortened, 
and the labor of calculation abridged. Tredgold proposed to 
make the deflection in proportion to the length of the beam, 
and the amount at the rate of one-fortieth of an inch (= 0'025 
inch) for every foot of length. He was undoubtedly right in 
the manner and probably so in the rate ; yet, as this is a mat- 
ter of opinion, it were better perhaps to leave the rate of de- 
flection open for the decision of those who use the rules, and 
thon it may be varied to suit the peculiarities of each case 



FEAMING-. 20 L 

that may arise. Any deflection within the limits of the elas- 
ticity of the material, may be given to beams used for some 
pm-poses, while others require to be restricted to that amount 
of deflection that shall not be perceptible to a casual observer. 
Let n represent, in the decimal of an incb, the rate of deflec- 
tion per foot of the length of the beam ; then the product of n^ 
multiplied by the number of feet contained in the length of 
the beam, will equal the total deflection, =: n I, Now, if ^ Z be 
substituted for p in the formulas, (26,) (27) and (28,) they will 
be rendered more available for general use. For example, let 
this substitution.be made in (26,) and there results — • 

where I is in feet, and 5, d and n in inches ; and for (27) — 
^ r w P w 
^=1^1 = ^' (30.) 

also for (28)— 

^ - 0-589 1)' n1 - 0-589 D' n' ^^^•) 
where the notation is as before, with also s and D in inches. 
In these formulas, w represents the weight in pounds concen- 
trated at the middle of the length of the beam. If the weight, 
instead thereof, is equcdly distributed over the length of the 
beam, then, since | of it concentrated at the middle will de- 
flect a beam to the same depth that the whole does when 
equally distributed, {Art. 281,) therefore— 



s n 



(33.) 



^== 0-589 1>'^' (3^-) 

wliere w equals the whole of the equally distributed load. 
Again, if the load is borne by more beams than one, laid 
parallel to each other— as, for example, a series or tier of floor 

26 



202 AMEEICAN HOUSE-CAEPENTEK. 

beams — and the load is equally distributed over the supported 
Burface or floor ; then, if /" represents the number of pounds ot 
the load contained on each square foot of the floor, or the 
pounds' weight per foot snperflcial, and g represents the dis- 
tance in feet between each two beams, or rather the distance 
from their centres, and I the length of the beam in feet, in the 
clear, between the supports at the ends ; then c I will equal 
the area of surface supported by one of the beams, and/* c I 
will represent the load borne by it, equally distributed over its 
length. Now, if this representation of the load be substituted 
for w in (32,) (33) and (3i) there results— 

J, if oi l' ifoV_ ,,^. 

^ - 0-589 i>* ^ - 0-589 i>* 7^ ^"^^'^ 



Practical Rules and Examples. 

323. — To ascertain the weight, placed upon the middle of a 
beam, that will cause a given deflection. 

Bule XX. — Multiply the area of the cross-section of the 
beam by the square of the depth and by the rate of the deflec- 
tion, all in inches ; multiply the product by the value of E^ 
Table II., and divide this product by the square of the length 
in feet, and the quotient w^ill be the weight in pounds required. 

Example. — What weight can be supported upon the middle 
• of a Georgia pine girder, ten feet long, eight inches broad, 
and ten inches deep, the deflection limited to three-tenths of 
an inch, or at the rate of 0"03 of an inch per foot of the length ? 
Here the area equals 8 X 10 = 80 ; the square of the depth 
equals 10 x 10 = 100 : 80 x 100 = 8,000 ; this by 0-03, the 
rate of deflection, the product is 240 ; and this by 2970, the 
value of ^for Georgia pine, Table II., equals 712,800. This 



FRAMING. 203 

product, divided by 100, the square of tlie length, the quotient, 
7,128, is the weight required in pounds. 

Rule XXL — Where the beam is square the weight may be 
found by the preceding rule or by this : — Multiply the square 
of the area of the cross-section by the rate of deflection, both 
in inches, and the product by the value of E^ Table 11. , and 
divide this product by the square of the length in feet, and the 
quotient will be the weight required in pounds. 

Examjjle. — What weight placed on the middle of a spruce 
beam will deflect it seven-tenths of an inch, the beam being 
20 feet long, 6 inches broad, and 6 inches deep? Here the 
area is 6 x 6 = 36, and its square is 36 X 36 = 1296 ; the rate 
of deflection is equal to the total deflection divided by the 

length, r^^rz. 0-035; therefore, 1296 X 0-035 = 45-36, and 

this by 1550, the value of E for spruce. Table IL, equals 
70,308. This, divided by 400, the square of the length, equals 
175*77, the required weight in pounds. 

Bule XXII. — When the beam is round find the weight by 
this rule: — Multiply the square of the diameter of the cross- 
section by the square of the diameter, and the product by the 
rate of deflection, all in inches, and this product by 0*589 
times tlie value of E^ Table II. This last product, divided by 
the square of the length in feet, will give the required weight 
in pounds. 

Exam^ple. — What weight on the middle of a round white 
pine beam will cause a deflection of 0*028 of an inch per foot, 
the beam being 10 inches diameter and 20 feet long? The 
square of the diameter equals 10 x 10 = 100 ; 100 x 100 — * 
10,000 ; this by the rate, 0*028, = 280, and this by 0*589 x 
1750, the value of ^, Table IL, for white pine, equals 288,610. 
This last product, divided by 400, the square of the length^ 
equals 721*5, the required weight m pounds. 

324.— To ascertain the weight that will produce a given de 



204 AMERICAN HOFSE-CAEPENTEE. 

flection, when the weight is equally distributed over the length 
oi' the beam. 

Bule XXIII. — ^The rules for this are the same as the three 
preceding rules, with this modification, viz., instead of the 
square of the length, divide by five-eighths of the square of 
the length. 

325. — In a series or tier of beams, to ascertain the weight 
per foot, equally distributed over the supported surface, that 
will cause a given deflection in the beam. 

Rule XXIY. — The rules for this are the same as Rules XX., 
XXL, and XXIL, with this modification, viz., instead of the 
square of the length, divide by the product of the distance 
apart in feet between each two beams, (measured from the 
centres of their breadths,) multiplied by five- eighths of the 
cube of the length, and the quotient will be the required 
weight in pounds that may be placed upon each superficial 
foot of the floor or other surface supported by the beams. In 
this and all the other rules, the weight of the material com- 
posing the beams, floor, and other parts of the constructions is 
understood to be a part of the load. Therefore from the ascer- 
tained weight deduct the weight of the framing, floor, plaster- 
ing, or other parts of the construction, and the remainder will 
be the neat load required. 

Exam/pie. — In a tier of white pine beams, 4 x 12 inches, 20 
feet long, placed 16 inches or 1§ feet from centres, w^hat 
weight per foot superficial may be equally distributed over 
the floor covering said beams — the rate of deflection to be not 
more than 0'025 of an inch per foot of the length of the beams. 
Proceeding by Rule XX. as above modified, the area of the 
cross-section, 4 x 12, equals 48 ; this by 144, the square of the 
depth, equals 6912, and this by 0*025, the rate of deflection, 
equals 172*8. Then this product, multiplied by 1Y50, the 
value of E, Table II., for white pine, equals 302,400. The 
distance between the centres of the beams is 1^ feet, "^he qube 



FRAMING. 



205 



of the length is 8,000, and ^ by I of 8,000 equals bfi66l The 
above 302,400, divided by 6,666^^, the quotient, 45-36, eqnak 
the required weight in pounds per foot superficial. The 
weight of beams, floor plank, cross-furring, and plastering oc- 
curring under every square foot of the surface of the floor, is 
now to be ascertained. Of the timber in every 16 inches by 
12 inches, there occurs 4 x 12 inches, one foot long ; this 
equals one-third of a cubic foot. Now, by proportion, if 16 
inches in width contains -J of a cubic foot, what will 12 inches 

J- X 12 12 

in width contain ? ^7^- = r, ttt = t\ =: i of sl cubic foot. 

lb 6 X lb 

The floor plank (Georgia pine) is 12 x 12 inches, and 1 J inches 

1- . - 

thick, equal to r-^ of a cubic foot, equals t^, equals -^\, Of the 

furring strips, 1x2 inches, placed 12 inches from centres, 
there will occur one of a foot long in every superficial foot. 
]^ow, since in a cubic foot there is 144 rods, one inch square 
and one foot long, therefore, this furring strip, 1 x 2 x 12 
inches, equals yf^ = yV ^^ ^ cubic foot. The weight of the 
timber and furring strips, being of white pine, may be esti- 
mated togetlier ; i -f ^\ = tI + tV = tI ^^ ^ cubic foot. White 
pine varies from 23 to 30 pounds. If it be taken at 30 pounds, 
the beam and furring together will weigh 30 X 4| pounds, 
equals 7'92 pounds. Georgia pine may be taken at 50 pounds 
per cubic foot ;'^^ the weight of the floor plank, then, is 50 X 
5^5- — 5*21 pounds. A superficial foot of lath and plastering 
will weigh about 10 lbs. Thus, the white pine, 7*92, Georgia 
pine, 5*21, and the plastering, 10, together equal 23*13 pounds ; 
this from 45-36, as before ascertained, leaves 22*23, say 22^ 
pounds, the neat weight per foot superficial that may be 
equally distributed over the floor as its load. 



* To get the weight of wood or any other material, multiply its specific gravity by 62'5, For Sp« 
ciHc Uraviues see Tables I., II., and III. and the Appendix for Weight of Materials. 



206 



AMERICAN HOIJSE-CARPENTEK. 



326. — To ascertain the weight when the beam is laid not 
horizontal, but inclined. 

Rule XXY. — In each of the foregoing rules, multiply the 
result there obtained by the length in feet, and divide the pro- 
duct by the horizontal distance between the supports in feet; 
and the quotient will be the required weight in pounds. 

The foregoing Rules, stated algebraically, are placed in the 
following table : — 



When the 


When the weight is 


When the beam is 


beam is laid 


Eect- 
angular. 


Square. 


Bound. 


Ilorizontal 


Concentrated at middle, w, in pounds, equals 
Equally di8triTt)uted, w, in pounds, equals 
By the foot superficial,^ in pounds, equals 
Concentrated at middle, w, in pounds, equals 
Equally distributed, w, in pounds, equals 
By the foot superficial,/ in pounds, equals 


(3S.) 
Enh d^ 


(39.) 

Ens^ 

I- 


(40.) 
•589 ^71 2)4 


Horizontal 


(41.) 

Enld^ 


(42.) 
Ens^ 


(43.) 

■UUEnB^ 

P 


norlzontal 


(44.) 

Enhd^ 

%cP 


(45.) 
En 8^ 

ye cP 


(46.) 

•UUEnDi 

cP 


Inclining 


(47.) 

Enl) d^ 
Ih 


(48.) 

En si 

Ik 


(49.) 

•m^EnDi 

Ih 


Inclining 


(50.) 
Enl)d^ 


(51.) 
Ens^ 


(52.) 

'UUEnD^ 

Ih 


Inclining 


(53.) 
Enld^ 
y^ cl h 


(54) 

En si 


(55.) 

•942 i^ 7? D 

cPh 



In the above table, h equals the breadth, and d equals the 
depth of cross-section of beam ; s equals the breadth of a side 
of a square beam, and D equals the diameter of a round 
beam ; n equals the rate of deflection per foot of the length ; 



TEAMING. 207 

/>, 5, 5, d and n^ all in inches ; I equals the length, c eqnals 
the distance between two parallel beams measured from the 
centres of their breadth ; K equals the horizontal distance 
between the supports of an inclined beam ; Z, g and A in feet ; 
w equals the weight in pounds on the beam ; f equals the 
weight upon each superficial foot of a floor or roof supported 
by two or more beams laid parallel and at equal distances 
apart ; ^E" is a constant, the value of which is found in Table 
II. ; T is any decimal, chosen at pleasure, in proportion to 
unity, as h is to c?, from which proportion h equals d r. 

327. — ^To ascertain the dimensions of the cross-section of a 
beam to support the required weight with a given deflection. 

Mule XXYI. — Preliminary. When the weight is concen- 
trated at the middle of the length. Multiply the weight in 
pounds by the square of the length in feet, and divide the pro- 
duct by the product of the rate of deflection multiplied by the 
value of ^., Table II., and the quotient equals a quantity which 
raay be represented by M- — referred to in succeeding rules. 

^ - if. (56.) 

Hule XXYII. — Preliminary. When the weight is equally 
distributed over the length. Multiply five-eighths of the weight 
in pounds by the square of the length in feet, and divide the 
product by the rate of deflection multiplied by the value of E^ 
Table II., and the quotient equals a quantity which may be 
represented by N- — referred to in succeeding rules. 

liule XXYIII. — Preliminary. When the weight is given 
per foot superficial and supported l)y two or more learns. 
Multiply the distance apart between two of the beams, (mea- 
sured from the centres of their breadth,) by the cube of the 
length, both in feet, and multiply the product by five-eighths 
of the weight per foot superficial ; divide this product by the 



208 AMEEICAN HOUSE-CAEPENTEE. 

product of the rate of deflection, multiplied by the value of E 
Table IL, and the quotient equals a quantity which may be 
represented by TI- — referred to in succeeding rules. 

^^ = U. (58.) 

Rule XXIX. — Preliminary. When the leaon is laid not 
horizontal, hut inclining. In Rules XX YI. and XXYII., 
instead of the square of the length multiply by the length, and 
by the horizontal distance between the supports, in feet. And 
in Eule XXYIIL, instead of the cube of the lengthy multiply 
by the square of the length, and by the horizontal distance 
between the supports, in feet. 



From (56) 
From (57) 
From (58) 



^ = iC. (59.) 

'^ = ^.. (60.) 

^-^=TJ,- (61.) 



Rule XXX. — When the heam is rectangular to find the 
dhnensions of the cross-section. Divide the quantity repre- 
sented by M, JV or U, (in preceding prehminary rules,) by 
the breadth in inches, and the cube root of the quotient will 
equal the required depth in inches. Or, divide the quantity 
represented by M, iV^or W, by the cube of the depth in inches, 
and the quotient will equal tlie required breadth in inches. 
Or, again, if it be desired to have the breadth and depth in 
proportion, as r is to unity, (where r equals any required deci- 
mal,) divide the quantity represented by 3f, JV or Z7, by the 
value of r, and extract the square root of the quotient : and 
the square root extracted the second time, will equal the depth 
in inches. Multiply the depth thus found by the value of r, 
and the product will equal the breadth in inches. 



FEAMIKG. 209 

Example. — To find the depth. A beam of spruce, laid on 
supports with a clear bearing of 20 feet, is required to support 
a load of 1674 pounds at the middle, and the deflection not to 
exceed 0*05 of an inch per foot ; what must be the depth when 
the breadth is 5 inches. By Kule XXYI. for load at middle : 
the product of 1674, the weight, by 400, the square of the 
length, equals 669,600. The product of 0-05, the rate of de- 
flection, multiplied by 1550, the value of E^ from Table IL, 
set opposite spruce, is 77-5. The aforesaid product, 669,600, 
divided by 77-5, equals 8640, the value of M. Then by Kule 
XXX., 8640, the value of if, divided by 5, the breadth, the 
quotient is 1728, and 12, the cube root of this, found in the 
table of the Appendix, equals the required depth in inches. 

Example. — To find the hreadth. Suppose that in the last 
example it were required to have the depth 13 inches ; in that 
case what must be the breadth ? The value of Jf, 8640, as 
just found, divided by 2197, the cube of the depth, equals 
8*9326, the required breadth — nearly 4 inches. 

Example. — To find loth hreadth and depths and in a certain 
proportion. Suppose, in the above example, that neither the 
breadth nor the depth are given, but that they are desired to 
be in proportion as 0*5 is to I'O. Now, having ascertained the 
value of Jf, by Kule XXYL, to be 8640, as above, then, by 
Kule XXX., 8640, divided by 0-5, the ratio, gives for quotient 
17,280. The square root of this (by the table in the Appen- 
dix,) is 131*45, and the square root of this square root is 11*465, 
the required depth. The breadth equals 11*465 x 0*5, which 
equals 5*7325. The depth and breadth may be 11^ by 5f 
inches. In cases where the load is equally disirihuted over 
the length of the beam, the process is precisely the same as set 
forth in the three preceding examples, except that five-eighth^^ 
of the weight is to be used in place of the whole weight ; anrl 
hence it would be a useless repetition to give examples to 
illustrate such cases. 

27 



210 AMERICAN HOrSE- CARPENTER. 

Example. — When the weight is per foot superficial to fim^d 
the depth. A floor is to be constructed to support 500 pounds 
on every superficial foot of its surface. The beams to be ot 
white pine, 16 feet long in the clear of the supports or walls, 
placed 16 inches apart, from centres, to be 4 inches] thick, and 
the amount oi diQ^QQiion not objectionable provided it be within 
the limits of elasticity. Proceeding by Rule XXYIIL, the 
product of 1-J feet, (equal to 16 inches,) multiplied by 4096, the 
cube of the length, equals 5461-|-. This, multiplied by 312*5, 
(equal to f of the weight,) equals 1,706,666. The largest rate 
of deflection within the limits of tlie elasticity of white pine is 
0-1022, as per Table 11. This, multiplied by 1750, the value of 
E for white pine. Table XL, equals 178'85. The former product, 
1,706,666, divided by the latter, 178-85, equals 9,542-5, the 
value of U. Now, by Rule XXX., this value of U, 9,542-5, 
divided by 4, the breadth, equals 2385-6, the cube root of 
which, 13*362, is the required depth— nearly 13f inches. 

Example. — To find the 'breadth. Suppose, in the last exam- 
ple, that the depth is known but not the breadth, and that the 
depth is to be 13 inches. Having found the value of Z7, as 
oefore, to be 9542-5, then by Rule XXX., dividing 9542*5, the 
value of TJ^ by 2197, the cube of the depth, gives a quotient of 
4-3434 and this equals the breadth — nearly 4f inches. 

Example. — To find the depth and hreadth in a given propor- 
tion. Suppose, in the above example, that the breadth and 
depth are both unknown, and that it is desired to have them 
in proportion as 0-7 is to 1-0. Having found the value of Z7, 
as before, to be 9542-5, then by Rule XXX., dividing 9542-5, 
the value of Z7, by 0*7, the quotient is 13,632, the square root 
of which is 116*75, and the square root of this is 10-805, the 
depth in inches. Then 10-805, multiplied by 0*7, the product, 
7*5635, is the breadth in inches. The size may be 7t6 ^J l^i^ 
inches. 

328.— Example, — In the case of inclined heims to find the 



FRAMING. 2xJ 

hxjth, A beam of white pine, 10 feet long in the clear of the 
bearino^, and laid at such an inclination that the hortzontal 
distance between the supports is 9 feet, is required to support 
12,000 pounds at the centre of its length, with the greatest 
allowable deflection within the limits of elasticity ; what must 
be its depth when its breadth is fixed at 6 inches ? By refer- 
ence to Table XL it is seen that the greatest value of ?^, within 
the limits of elasticity, is 0-1022. By Rule XXYL, for con- 
centrated load, and Rule XXIX., for inclined beams, 12,000, 
the weight, multiplied by 10, the length, and by 9, the liori- 
zontal distance, equals 1,080,000. The product of 0*1022, the 
greatest rate of deflection, by 1750, the value of E, Table IL, 
for white pine, equals 178-85. Dividing 1,080,000 by 178-85, 
the quotient is 6038*58, the value of M. Now, by Rule 
XXX., for rectangular beams, 6038*58, the value of Jf, divid- 
ed by 6. the breadth, the quotient is 1006*43. The cube root 
of this, 10'02, a trifle over 10 inches, is the depth required. 

Example. — In case of inclined teams to find the treadth. 
In the last example suppose the depth fixed at 12 inches; 
then by Rule XXX., 6038*58, the value of J[f, as above found, 
divided by 1728, the cube of the depth, equals 3-4945, or 
nearly 3|- inches — the breadth required. 

Example. — Again, in case the hreadth and depth are to he in 
a certain proportion j as, for example, as 0*4 is to unity. 
Then by Rule XXX., 6038'58, the value of J[f, found as above, 
divided by 0*4, equals 15,096*45, the square root of which is 
122*87, and the square root of this square root is 11*0843, a 
trifle over 11 inches — the depth required. Again, 11 multi- 
plied by the decimal 0*4, (as above,) equals 4*4, a little over 
41 inches — the breadth required. 

In the three preceding examples, the weight is understood 
to be concentrated at the middle. If, however, the weight 
had been equally distributed, the same process would have 
been used to obtain the dimensions of the cross-section, with 



212 AMERICAN HOTJSE-CAKPENTEE. 

only one exception ; viz. f of tlie weight instead of the whole 
weight would have been used. (See Rule XXYIL) 

Example. — In case of inclined learns; the weight per foot 
superficial^ and home hy two or more heams. A tier of spruce 
beams, laid with a clear bearing of 10 feet, and at 20 inches 
apart from centres, and laid so inclining that the horizontal 
distance between bearings is 8 feet, are required to sustain 40 
pounds per superficial foot, with a deflection not to exceed 
02 inch per foot of the length ; what must be the depth 
when the breadth is 3 inches ? Proceeding by Rule XXIX. 
for inclined beams, and hj Rule XXYIIL, If, {■= 20 inches,) 
the distance from centres, multiplied by 100, the square of the 
length, and by 8, the horizontal distance between bearings, 
equals 1,333J ; this, by f x 40, five-eighths of the weight, 
equals 33,333^. This, divided by 0-02 x 1550, the rate of 
deflection, by the value of E^ Table IL, for spruce, equal to 
31, equals 1075-27, the value of U. Now by Rule XXX. for 
rectangular beams, 1075*27, divided by 3, the breadth, equals 
358*42, the cube root of which, 7*1, is the required depth in 
inches. 

Example. — The same as the preceding ; hut to find the 
hreadth^ when the depth is fixed at 8 inches. By Rule XXX., 
1075*27, the value of U, divided by 512, the cube of the 
depth, equals 2*1 — the breadth required in inches. 

Example. — The same as the next hut one preceding ; hut to 
find the hreadth and depth in the proportion of O'Z to 1*0, or 
3 to 10. By Rule XXX., 1075*27, the value of U, divided by 
0*3, the value of r^ equals 3584*23. The square root of this is 
59*869, and the square root of this square root is 7*737 — the 
depth required in inches. This 7*737, multiplied by 0*3, the 
value of r, equals 2*3211 — the required breadth in inches. 
The dimensions may, therefore, be 2/^ by 7f inches. 

I^ule XXXI. — When the heam is square to find the side 
Extract the square root of the quantity represented by J/", iV 



FRAMING. 213 

or U, in preliminary Kules XXYI., XXYII. and XXYIII., 
and the square root of this square root will eqnal the side 
required. 

Example. — A beam of chestnut, having a clear bearing of 8 
feet, is required to snstain at the middle a load of 1500 
pounds ; what must be tlie size of its sides in order that the 
deflection shall not exceed 0*03 inch per foot of its length ? By 
Rule XXYI., 1500, the load, multiplied by 64, the square of 
the length, equals 96,000. This product divided by 0-03 times 
2330, the value of E^ Table II., for chestnut, gives a quotient 
of 13734, the quantity represented by M. E'ow by Rule 
XXXL, the square root of 1373-4: is 37*05, and the square 
root of this is 6 '087. The beam must, therefore, be 6 inches 
square. In this example, had the load, instead of being con- 
centrated at the middle, been equally distributed over its 
length, the side would have been equal to the side just found, 
multiplied by the fourth root of f or of 0*625, equal to 6*087 
X 0*889 = 5*4 inches. (See Rules XXYII. and XXXL) 

Example, — In the case where the weight is per foot superfi- 
cial and home hy two or more heams. A floor, the beams of 
which are of oak, and placed 20 inches or 1§ feet apart from 
centres, and which have a clear bearing of 20 feet, is required 
to sustain 200 pounds per superficial foot, the deflection not to 
exceed 0*025 inch per foot of the length, and the beam to be 
square. By Rule XXYIII., If, the distance from centres, 
multiplied by 8000, the cube of the length, equals 13,333^ ; 
and this by 125, (being I of 200 pounds,) equals IfiQQ.mQl. 
Dividing this by 0*025 times 2520, the value of E, Table II., 
for oak, the quotient is 26,455 — a number represented by U. 
E'ow by Rule XXXL, the square root of this number is 162*65. 
and the square root of this square root is 12*753 — the required 
side. The beam may be 12f inches square. 

Example. — Inclined square heams^ load at middle. A bar 
of cast-iron, 6 feet long in the clear of bearings, and laid 



214: AMERICA]^ HOTJSE-CAEPENTER. 

inclining so that the horizontal distance between tlie bearings 
is 5 feet, is required to sustain at the middle 3000 pounds, and 
the deflection not to exceed 001 inch per foot of its length ; 
what must be the size of its sides ? 

By Rule XXYI. for load at middle, modified by Rule 
XXIX. for inclined beams ; 3000, the weight, multiph'ed by 6, 
the length, and by 5, the horizontal distance between bear- 
ings, equals 90,000. The rate of deflection, 0-01, by 30,500, 
the value of E^ Table 11., for cast-iron, equals 305 ; and 9000 
divided by 305, equals 295*082, the value of M. Now by Rule 
XXXI. for square beams, the square root of 295-082 is 1T"18, 
the square root of which is 4*115 — the size of the side required ; 
a trifle over 4i, the bar may, therefore, be 4|- inches square. 

Example. — Same as preceding^ hut the weighs equally distT^' 
luted. By Rule XXYII. f of the weight is to be used instead 
of the weight ; therefore 295*082, the value of J/, as above, 
multiplied by I, will equal 181*426, the value of iT. By Rule 
XXXI. the square root of 184*426 is 13*58, the square root of 
which is 3*685 — the size of the side required ; equal to nearly 
3}^ inches square. 

Example. — Same as preceding case^ hut the weight per fool 
superficial.^ and sustained by 2 or more bars, placed 2 feet 
from centres, the load being 250 pounds per foot superficial. 
By Rule XXYIII., modified by Rule XXIX., the distance 
from centres, 2, multiplied by 36, the square of the length, 
and by 5, the horizontal distance, equals 360. This by 156-25, 
five-eighths of the weight, equals 56,250. The rate of deflec- 
tion, 0*01, by 30,500, the value of ^, Table IL, for cast-iron, 
equals 305. The above 56,250, divided by 305, equals 184*426, 
the value of U. ]^ow by Rule XXXI. the square root of 
184*426, the value of Z7, is 13*58, the square root of which is 
3*685 — the size of the side required. It will be observed that 
this result is precisely like that in the last example. This is as 
it should be, for each beam has to sustain the weight on 2 x 6 



FEAMING. 215 

ir 12 superficial feet, equal to 12 x 250, equal 3(00 pounds; 
and all the otlier conditions are parallel. 

Rule XXXII. — When the hearn is round to find the diame- 
ter. Divide the value of Jf, JST ov U, found by Rules XXYI., 
XXYII. or XXYIIL, by the decimal 0-589, and extract the 
square root: and the square root of this square root will be 
the diameter required. 

Example. — In the case of a concentrated load at middle A 
round bar of American iron, of 5 feet clear bearing, is required 
to sustain 800 pounds at the middle, with a deflection not to 
exceed 0*02 inch per foot ; what must be its diameter ? By 
Kule XXYI. for load at middle, 800, the weight, multiplied 
by 25, the square of the length, equals 20,000. The rate of 
deflection, 0*02, by 51,400, the value of ^, Table II., for Ame- 
rican wrought iron, equals 1028. The above 20,000, divided 
by 1028, equals 19*4:552, the value of M. Now, by Rule 
XXXIL, 19-1:552, the value of Jf, divided by 0-589 equals 
33-03, the square root of which is 5*71:7, and the square root 
of this is 2-397, nearly 2-4, the diameter required in inches, 
equal to 2f large. 

Example. — Same case as the preceding^ hut the load equally 
distributed. By Rule XXYII., flve-eighths of the weight is to 
be used instead of the whole weight ; therefore the above 
33-03, multiplied by I, equals 20-64375, the square root of 
which is 4-544, and the square root of this square root is 2-132, 
the diameter required in inches, 2|- inches large. 

Example. — When the weight is per foot superficial^ and sus- 
tained ly two or more bars or beams. The conditions being 
the same as in the preceding examples, but the weight, 100 
pounds per foot, is to be sustained on a series of round rods, 
placed 18 inches apart from centres, equal 1*5 feet. By Rule 
XXYIIL, for weight per foot superficial, 1*5, the distance 
from centres, multiplied by 125, the cube of the length, and 
by 62*5, five-eighths of the weight, equals 11,718*75. This 



216 AMERICAN HOUSE-CAEPENTEE. 

divided by 1028, tlie product of the rate of deflection by the 
value of E^ as found in the preceding example, equals 114, 
the value of TI. I^ow by Eule XXYII., 114, the value of U, 
divided by 0*589, equals 1942, the square root of which is 
4407, and the square root of this square root is 2*099, the 
diameter required — very nearly 2yV inches. 

Examjple. — When the heam is round and laid inclining^ the 
weight concentrated at the middle. A round beam of white 
pine, 20 feet long between bearings, and laid inclining so that 
the horizontal distance between bearings is 18 feet, is required 
to support 1250 pounds at the middle, with a deflection not to 
exceed 0*05 inch per foot ; what must be its diameter ? By 
Eule XXYL for load at middle, modified* by Eule XXIX. for 
inclined beams, 1250, the weight, multiplied by 20, the length, 
and by 18, the horizontal distance, equals 450,000. The rate 
of deflection, 0'05, multiplied by 1750, the value of E^ Table 
II., for white pine, equals 87*5. The above 450,000 divided 
by 87-5, equals 5142-86, the value of M. JSTow by Eule 
XXXII. for round beams, 5142'86, the value of Jf, divided by 
0'589, equals 8731*5, the square root of which is 93*44, and 
the square root of this square root is 9*667, the diameter re- 
quired — equal to 9§ incbes. 

Examjple. — Same as in preceding examjple ^ hut the weight 
equally distributed. By Eule XXYII., five-eighths of the 
weight is to be used instead of the whole weight, therefore 
8731*5, the result in the last example just previous to taking 
the square root, multiplied by f , equals 5457*2, the square root 
of which is 73*87, and the square root of this square root is 
8*59, the diameter required — nearly 8f inches. 

Example. — Same as in the next hut one preceding example^ 
hut the weight per foot superficial^ and supported hy two or 
more heams. A series of round hemlock poles or beams, 10 
feet long clear bearing, laid inclining so as that the horizontal 
distance between the supports equals 7 feet, and laid 2 feet 



FRAMING. 317 

and 6 inches apart from centres, are required to support 20 
ponnds per superficial foot without regard to the amount of 
deflection, provided that the elasticity of the material be not 
injured; what must be tlieir diameter? By Rule XXYIII. 
for weight per foot superficial, modified by Rule XXIX. for 
inclined beams, 2*5, the distance from centres, inultiplied by 
100, the square of the length, and by 7, the horizontal distance 
between bearings, and by five-eighths of the weight, 12*5; 
equals 21,875. The greatest value of n^ Table IL, for hem- 
lock, 0-08794, multiplied by 1210, the value of E, Table IL, 
for hemlock, equals 109-01:56. The above 21,875, divided by 
109-0156, equals 200*6, the value of V. IsTow by Rule 
XXXIL, the abofe 200-6, divided by 0-589, equals 340-6, the 
square root of which is 18-4:6, and the square root of this 
square root is 4-296, the diameter required — equal to 4y\ 
inches nearly. 

329. — The greater the depth of a beam in proportion to the 
thickness, the greater the strength. But when the difference 
between the depth and the breadth is great, the beam must be 
stayed, (as at Fig. 228,) to prevent its falling over and break- 
ing sideways. Their shrinking is another objection to deep 
beams ; but where these evils can be remedied, the advantage 
of increasing the depth is considerable. The following rule is, 
to find the strongest form for a hemn out of a given quantity 
of timher. 

Rule. — Multiply the length in feet by the decimal, 0-6, and 
divide the given area in inches by the product ; and the 
squai'e of the quotient will give the depth in inches. 

Example. — What is the strongest form for a beam whose 
given area of section is 48 inches, and length of bearing 20 
feet ? The length in feet, 20, multiplied by the decimal, 0*6, 
gives 12 ; the given area in inches, 48, divided by 12, gives a 
quotient of 4, the square of which is 16 — this is tlie depth in 
inches ; and the breadth must be 3 inches. A beam 16 inches 

28 



218 



AMERICAN HOUSE-CAEPENTEE. 



by 3 would bear twice as much as a square beam of the same 
area of section ; which show^s how important it is to make 
beams deep and thin. In many old buildings, and even in 
new ones, in country places, the very reverse of this has been 
practised ; the principal beams being oftener laid on the 
broad side than on the narrower one. 

The foregoing rules, stated algebraically, are placed in the 
following table. 

TABLE V. STIFFNESS OF BEAMS I DIMENSIONS. 



When 

the 

beam 

is laid 


When the weight is 


Eectangular. 


Square. 


Bound. 


Value of 
depth. 


Value of 
breadth. 


When?>=^/', 
value of d. 


Value of a 
side. 


Value of th« 
diameter 




Concentrated at middle 
Equally distributed 
By the foot superficial 


(62.) 
\/ wP 
Enb 


(63.) 
w P 


(64.) 
i^wP 
Enr 


(65.) 
UwP 
^ En 


(66.) 
\/ wP 




End^ 


■5S9 En 


1 


(67.) 
^ Enh 


(68.) 
% w P 
End^ 


(69.) 

i^% w p 

Enr 


(70.) 
t/% w P 
En 


(71.) 
^9424 En 




(72.) 

Enb 


(73.) 
VsfcP 
End^ 


(74.) 
Enr 


(75.) 

i/%feP 

En 


(76.) 
V fcP 
•9424 En 




Concentrated at middle 
Equally distributed 
By the foot superficial 


(77.) 
l/wlh_ 


(78.) 
wlh 
End- 


(79.) 
Enr 


(80.) 


(81.) 
^ '5S9 En 


bO 


(82.) 
^-Er,b~ 


(83.) 
%wlh 
End^ 


(84.) 

\/VsWlh 

Enr 


(85.) 
En 


(86.) 

4/ Wlh 

■9424 En 




(87.) 
Enh 


(88.) 
%fcPh 
Endi 


(89.) 


(90.) 


(91.) 
\/ fcPh_ 
■9424 En 



In the above table, h equals the breadth, and d the depth of 
cross-section of beam ; n equals the rate of deflection per foot of 
the length ; h, d and n, all in inches. Also, I equals the lengthy 
c the distance between two parallel beams measured from the 



FRAMING. 



219 



centres of their breadth, and A equals the horizontal distance 
between the supports of an inclined beam ; Z, c and A, all in 
feet. Again, %o equals the weight on the beam,/" equals the 
weight upon each superficial foot of a floor or roof, supported 
by two or more beams laid parallel and at equal distances 
apart ; w and f in pounds. And r is any decimal, chosen at 
pleasure, in proportion to unity, as 1) is to d — from which pro- 
portion h = d r. ^ is a constant the value of which is found 
in Table II. 

330. — To ascertain the scantling of the stiff est team that can 
he cut from a cylinder. Let d a ch^ ^Fig. 223,) be the section, 
and e the centre, of a given cylinder. Draw the diameter, 
ah 'j upon a and Z>, with the radius of the section, describe tlie 
arcs, d e and e c ', join d and a^ a and <?, c and J, and h and <:?/ 
then the rectangle, d a c h^ will be a section of the beam 
required. 




Fig. 



331. — Resistance to Rupture. — ^The resistance to deflectimi 
having been treated of in the preceding articles, it now re- 
mains to speak of the other branch of resistance to cross 
strains, namely, the resistance to rupture. When a beam is 
laid horizontally and supported at each end, its strength to resist 
a cross strain, caused by a weight or vertical pressure at the 
middle of its length, is directly as the breadth and square of 
the depth and inversely as the length. If the beam is square 
or the depth equal to the breadth, then the strength is directlj 



220 AMERICAN irOUSE-CARPENTEE. 

as the cube of a side of the beam and inversely as the length, 
and if the beam is round the strength is directly as the cube 
of the d^'ameter and inversely as the length. 

When the weight is concentrated at any point in the length, 
the strength of the beam is directly as the length, breadth, and 
square of the depth, and inversely as the product of the two 
parts into which the length is divided by the point at which 
the weight is located. 

When the beam is laid not horizontal but inclining, the 
strength is the same as in each case above stated, and also in 
proportion, inversely as the cosine of the angle of inclination 
with the horizon, or, which is the same thing, directly as the 
length and inversely as the horizontal distance between the 
points of support. 

When the weight is equally diffused over the length of a 
beam, it will sustain just twice the weight that could be sus- 
tained at the middle of its length. 

A beam secured at one end only, will sustain at the other 
end just one-quarter of the weight that could be sustained at 
its middle were the beam supported at eaeh end. 

These relations between the strain and the strength exist in 
all materials. For any particular kind of material, 

^, = S; (92.) 

S, representing a constant quantity for all materials of like 
strength. The superior strength of one kind of material over 
another is ascertained by experiment ; the value of jS being 
ascertained by a substitution of the dimensions of the piece tried 
for the symbols in the above formula. Having thus obtained 
the value of xS, the formula, by proper inversion, becomes use- 
ful in ascertaining the dimensions of a beam that will require 
a certain weight to break it ; or to ascertain the weight that 
will be required to break a certain beam. It will be observed 
in the preceding formula, that if each of the dimensions of the 



TEAMING. 



221 



beam equal unity, then w = S. Hence, S is equal to the 
weight required to break a beam one inch square and one 
foot long. The values of S, for various materials, have been 
ascertained from experiment, and are here recorded : — 



TABLE VI. STRENGTH. 



Materials. 



Green plate-glass 
Spruce .... 
Hemlock .... 
Soft white pine 
Hard white pine 
Ohio yellow pine 
Chestnut .... 
Georgia pine 

Oak 

Locust .... 
Oast-iron (from 1550 to 2280) 



Value ofS. 


Number of 
Experiments. 


178 


4 


345 


5 


363 


7 


890 


9 


449 


1 


454 


2 


503 


2 


510 


7 


574 


2 


742 


2 


1926 


29 



The specimens broken were of various dimensions, from one 
foot long to three feet, and from one inch square to one by 
three inches. The cast-iron specimens were of the various 
kinds of iron used in this country in the mechanic arts. S 
may be taken at 2,000 for a good quality of cast-iron. It; is 
usual in determining the dimensions of a beam to suppose it 
capable of sustaining safely one-third of the breaking weight, 
and yet Tredgold asserts that one-fifth of tie breaking weight 
will in time injure the beam so as to give it a permanent set 
or bend, and ITodgkinson says that cast-iron is injured by any 
weight however small, or, in other words, that it has no elastic 
power. However this may be, experience has proved cast- 
iron quite reliable in sustaining safely immense weights for 
a long period. Practice has shown that beams will sustain 
safely from one-third to one-sixth of their breaking weight. 
If the load is bid on quietly, and is to remain where laid, at 
rest, beams may be trusted with one-third of their breaking 
weight, but if the load is moveable, or subject to vibration. 



222 AMERICAN HOTJSE-CAEPENTER 

one-quarter, one-fifth, or even, in some cases, one-sixtli is quite 
a sufficient proportion of the breaking load. 

332. — ^The dimensions of beams should be ascertained only 
by means of the rules for the stiffness of materials, (Arts. 320; 
323, et seq.,) as these rules show more accurately the amount 
of pressure the material is capable of sustaining without injury. 
Yet owing to the fact that the rules for the strength of mate- 
rials are somewhat shorter, they are more frequently used 
than those for the stiffness of materials. In order that the 
proportion of the breaking weight may be adjusted to suit cir- 
cumstances it is well to introduce into tlie formula a symbol 
to represent it. The proportion represented by the symbol 
may then be varied at discretion. Let this symbol be a, a 
decimal in proportion to unity as the safe load is to the break- 
ing load, then S a will equal the safe load. Hence, 

w — ^ (93.) 

for a safe load at middle on a horizontal beam supported at 
both ends ; and 

w — -J (94.) 

for a safe load equally diffused over the length of the beam : 
and 

for the load, per superficial foot, that can be sustained safely 
upon a floor supported by two or more beams, c being the dis- 
tance in feet from centres between each two. beams, and/" the 
load in pounds per superficial foot of the floor. Generally, in 
(93,) (94,) and (95,) w equals the load in pounds ; S^ a constant, 
the value of which is found in Table YI. ; a a decimal, in pro- 
portion to unity as the safe load is to the breaking load ; I the 
length in feet between the bearings ; and h and d the breadth 
and depth in inches. 



FRAMING. 



223 



TO FIND THE WEIGET. 

333. — The formulas for ascertaining the weight in tho seve- 
ral cases are arranged in the following table, where c, /, w^ S^ 
a, I, h and d represent as above ; and also s equals a side of 
a square beam ; D equals the diameter of a cylindrical beam : 
m and n equal respectively the two parts into which the length 
is divided by the point at which the weight is located ; and h 
equals the horizontal distance between the supports of an 
inclined beam. 



TABLE Vn. STRENGTH OF BEAMS ; SAFE WEIGHT. 



When the 


Wlien the weight is 


When the beam is 


Keara is laid 


Ecctantrular. 


Square. 


Eound. 




Concentrated at middle, w, in 
pounds, equals 

Equally distributed, w, in 
pounds, equals 

By the foot superficial, f, in 
pounds, equals 

Concentrated at any point in 
the length, w, in pounds, equals 


(96.) 

8al)d;^ 

I 


(97.) 

Sas^ 

I 


(98.) 

•589i)3^a 

I 


3 


(99.) 
I 


(100.) 


(101.) 

1-178 X>3^ a 

I 


(102.) 


(103.) 

2Sas^ 

cl 


(104.) 

1-178 i>3^ a 

cP 




(105.) 

Sald^^l 

4:inn 


(106.) 
Sals^ 
4.mn 


(107.) 

'UimSal 

m n 




Concentrated at middle, w, in 
pounds, equals 

Equally distributed, to, in 
pounds, equals 

By the foot superficial, f, in 
pounds, equals 

Concentrated at any point In 
the length, w, in pounds, equals 


(108.) 

Sadd^ 

h 


(109.) 
h 


(110.) 
■589 2>^^ a 




h 


1 


(111.) 
2 ^a 5(^2 
A ■ 


(112.) 
h 


(119.) 

1178 D^S a 

h 


(114.) 
2Sabd-i 
■ chl ■ 


(115.) 

^8(18^ 

chJ- 


(116.) 
cM 




(117.) 

Sahd'i p 

4:hmn 


(118.) 
Salis^ 
Ahmn 


(119.) 

•147 i>*^a^ 

hmn 






224 iJslEEIQiLN' HOIISE-CAEPENTEE, 

1/ 



i . 



J Practical Hules and Exanvples. 

Hide XXXIII. — To find the weight that may be supported 

safely at the middle of a beam laid horizontally. Multiply 

the value of 8^ Table YI., by a decimal that is in proi3ortion 

to nnity as the safe weight is to the breaking weight, and 

divide the product by the length in feet. Then, if the beam 

is rectangular, multiply this quotient by the breadth and by 

i(^ the square of the depth, and the product will be the required 

i^ \^ weight in pounds ; or, if the beam is square, multiply the said 

quotient, instead, by the cube of a side of the beam and the 

) 5 product will be the required weight in pounds ; but, if the 

beam is round, multiply the aforesaid quotient, instead, by 

'/ '589 times the cube of the diameter, and the product will be 

<^ the required weight in pounds. 

Example. — What weight will a rectangular white pine 
beam, 20 feet long, and 3 by 10 inches, sustain safely at the 
middle, the portion of the breaking weight allowable being 
0*3? By the above rule, 390, the value of S for white pine, 
Table YI., multiplied by 0-3, the decimal referred to, equals 
)l 117, and this divided by 20, the length, the quotient is 5*85. 

f '' Now the beam being rectangular, this quotient multiplied by 

3 and by 100, the breadth and the square of the depth, the 
product, 1T55, is the desired weight in pounds. 

Example. — If the above beam had been square^ and 6 by 6 
inches, then the quotient, 5'85, multiplied by 216, the cube of 
6, a side, the product, 1263*6, is the weight required in pounds. 
Example. — If the above beam had been round^ and 6 inches 
diameter, then the above quotient, 5*85, multiplied by '589 
times 216, the cube of the diameter, the product, YM*26, would 
be the required weight in pounds. 

Rule XXXIY. — To find the weight that may be supported 
safely when equally distributed over the length of a beam, 
laic^ fwrizontally. Multiply the result obtained, by Eule 

/ vy 



^^ 





2 2 ''' 

XXXIIL, "by 2, and the product will be the reqmred weight 
in pounds. 

Exam.jple. — In the example, under Rule XXXIIL, the safe 
weight at middle of rectangular beam is found to be 1755 
pounds. This multiplied by 2, the product, 3510, is the weight 
the beam will bear safely if equally distributed over its length. 

Example. — So in the case of the square beam, 2527-2 pounds 
is the weight, equally distributed, that may be safely sus- 
tained. 

Exanvple. — And for the round beam 1488-52 is the required / 
weight. 

Bule XXXY. — ^To ascertain the weight per superficial foot 
that may be safely sustained on a floor resting on two or more i 
beams laid horizontally and parallel. Multiply twice the value 
of S^ Table YI., by the decimal that is in proportion to unity, 
as the safe weight is to the breaking weight, and divide tl e 
product by the square of the length, in feet, multiplied by the 
distance apart, in feet, between the beams measured from their 
centres. E"ow, if the beams are rectangular^ multiply this 
quotient by the breadth and by the square of the depth, both 
in inches, and the product will be the required weight in 
pounds; or if the beams are square^ multiply said quotient, 
instead, by the cube of a side of a beam and the product will 
be the required weight in pounds. But if the beams ai-e 
rounds multiply the aforesaid quotient, instead, by -589 times 
the cube of the diameter, and the product will be the weight 
required in pounds. 

Example. — "What weight may be safely sustained on each 
foot superficial of a floor resting on spruce beams, 10 feet long, 
3 by 9 inches, placed 16 inches, or 1^ feet, from centres: the 
portion of the breaking weight allowable being 0*25 ? By the 
Rule, 690, twice the value of spruce, Table YL, multiplied by 
0-25, the decimal aforesaid, equals 172-5. This product divided 
by 100, the square of the length, multiplied by 1|-, the distance 

29 






71. rr-i i: 



^-^ 



226 AMERICAN HOUSE-CAKPENTEE. 

from centres, equals 1-294. 'Now this quotient multiplied by 
3, the breadth, and by 81, the square of the depth, the product 
314-44 is the required weight in pounds. 

Had these beams been square, and 6 by 6 inches, the re- 
quired weight would be 279-5 pounds. 

Or, if round, and 6 inches diameter, 164-63 pounds. 

J2ule XXXYI. — To ascertain the weight that may be sus- 
tained safely on a beam when concentrated at any point of its 
length. Multiply the value of S, Table YI., by the decimal in 
proportion to unity, as the safe weight is to the breaking 
weight, and by the length in feet, and divide the product by 
four times the product of the two parts, in feet, into which the 
length is divided, by the point at which the weight is concen- 
trated. Then, if the beam is rectangular, multiply this quo- 
tient by the breadth and by the cube of the depth, both in 
inches, and the product will be the required weight in pounds. 
Or, if the beam is square, multiply the said quotient, instead, 
by the cube of a side of the beam, and the product wiSl be the 
required weight in pounds. But if the beam is round, multi- 
ply the aforesaid quotient by '589 times the cube of the dia- 
meter, and the product will be the weight required. 

Example. — What weight may be safely supported on a 
Georgia pine beam, 5 by 12 inches, and 20 feet long ; the weight 
plaeed at 5 feet from one end, and the proportion of the break- 
ing weight allowable being 0*2 ? By the rule, 510, the value of 
S for Georgia pine. Table YL, multiplied by 0-2, the decimal 
referred to, equals 102 ; this by 20, the length, equals 2040 ; 
this divided by 300, (= 4 X 5 x 15,) or 4 times the product of 
the two parts into which the length is divided by the point at 
which the weight is located, equals 6*8. The beam being rect- 
a/ngular, this quotient multiplied by 5, the breadth, and bj 
144, the square of the depth, equals 4896, the required weight. 

A beam, 8 inches square, other conditions being the same as 
in the preceding case, would sustain safely 3481*6 pounds. 



FRAMING. 227 

And a round beam, 8 inches diameter, will sustain safelj, 
under like conditions, 2050*66 pounds. 

Rule XXXYIL— To find the weight that may be safely 
sustained on inclined heams. Multiply the result found for 
liorizontal beams in preceding rules, by the length, in feet, 
and divide the product by the horizontal distance between the 
supports, in feet, and the quotient will be the required weight. 

Example. — What weight may be safely sustained at the 
middle of an oak beam, 6 X 10 inches, and 10 feet long, (set 
inclining, so that the horizontal distance between the supports 
is 8 feet,) the portion of the breaking weight allowable being 
0*3 ? The result for a horizontal beam, by Rule XXXIII., is 
10,332 pounds. This, multiplied by 10, the length, and divid- 
ed by 8, the horizontal distance, equals 12,915 pounds, the 
required weight. 

TO FIND THE DIMENSIONS. 

334. — ^The following table exhibits, algebraically, rules for 
ascertaining the dimensions of beams required to support 
given weights ; where 5 equals the breadth, and d the depth 
of a rectangular beam, in inches ; I the length between sup- 
ports; A the horizontal distance between the supports of an 
inclined beam, and c the distance apart of two parallel beams, 
measured from the centres of their breadth, ?, A, and c, in feet ; 
w equals the weight on a beam ; f the weight on each super- 
ficial foot of a floor resting on two or more parallel beams ; R 
equals a load on a beam, and m s^nd n the distances, respect- 
ively, at which R is located from the two supports ; also jP is 
a weight, and g and k the distances, respectively, at which P 
is located from the two supports ; also m-{-n = l = g-{-Jc^' w, 
y, R, and P, all in pounds ; m, n, g, and ^, in feet. /S' is a 
constant, the value of which is found in Table YI. ; <^ is a 
decimal in proportion to unity as the safe load is to the break 



228 



AMERICAN HOUSE-CARPENTEE. 



ing load ; t* is a decimal in proportion to nnitj as 5 is to c? / 
from which h =d r ; a and r to be chosen at discretion. 



TABLE Vin. — STRENGTH 



When the 


"WTien the 
weight is 


Rectangular. 


Deam is laid 


Value of depth. 


Value of breadth. 




Concentrated at 
middle 

Equally distribut- 
ed 

By the foot super- 
ficial 

Concentrated at 
any point in the 
length 

At two or more 
points in the 
length 


(120.) 

Birr 


(121.) 
wl 

Sad^ 




(125.) 
4/ wl 
2Sab 


(126.) 

wl 

TS^^'i 


1 

(^ 


(130.) 
^2Sab 


(181.) 

fcP 

2Sadi 




(135.) 
A/4:Wmn 


(186.) 
4:wmn 




Sabl 


Sada 




(140.) 
/^4:{Bmn + Pg'k + <Sic.) 
Sabl 


(141.) 
SadAl 




Concentrated at 
middle 

Equally distribut- 
ed 

By the foot super- 
ficial 

Concentrated at 
any point in the 
length 

At two or more 
points in the 
length 


(145.) 
Sab 


(146.) 
wh 

'Jad^ 




(150.) 
^ wh 
2Sab 


(151.) 
wh 

2Sa<P 


1 


(165.) 
Vfchl 
2Sab 


(156.) 
fchl 
2Sad2 




(160.) 
A/4^hwmn 

-s~5rb P 


(161.) 
4:hwmn 
Sad;^P 




(165.) 

A/4:h{^Bmn->rPgh + &e.) 

Sabt^ 


(166.) 
^h{Rmn-itPgh + dbe.) 

SadH^ 



FEAMING. 



OF BEAMS ; DIMENSIONS. 





Square. 


Bound. 


When h ^dr. value of d. 


Value of a side. 


Value of the diameter. 


(122.) 
Sar 


(123.) 

X/wl 

Sa 


(124.) 

•589 >S a 


(127.) 

V ^^ 
2Sar 


(128.) 
2Sa 


(129.) 
3/ wl 

l-H%Sa 


(132.) 
2Sar 


(133.) 
2ASa 


(134.) 
1-178 «S a 


(137.) 
Sari 


(138.) 
'Sai^ 


(139.) 
3/ 'jcmw. 
•147xSaZ 


(142.) 

U^{Emn + Pgk + <&g.) 

Sari 


(143.) 


(144.) 

X/(,B mn + Pgk + <fec.) 

•147 aS a i 


(147.) 
Sar 


(148.) 
Sa 


(149.) 

•589 ^ a 


(152.) 
2Sar 


(153.) 

2Sa 


(154.) 
1-178^ a 


(157.) 
2AS'a/' 


(158.) 
2^a 


(159.) 

\/ fehl 
VllQSa 


(162.) 
8/4 A w m ?t 


(163.) 
8/4 hwmn 

Salf 


(164.) 

Ij htomn 
•U7 SaT' 


(167.) 
Sarh 


(168.) 

8/4 A (i? W 71 + P ^ ^ + <«C.) 


(169.) 

8/^ (Pm7i + Pe'A; + <fec.) 

'UTS a I' 



230 AMERICAN HOUSE-CAEPENTEK. 

Practical Rules and Exarajples. 

Rule XXXYIII. — Preliminary. When the weight is oon 
centrated at the middle. Multiply the weig'lit, in pounds, by 
the length, in feet, and divide the product by the vahie of S^ 
Table YL, multiplied by a decimal that is in proportion to 
unity as the safe weight is to the breaking weight, and the 
quotient is a quantity which may be represented by J, referred 
to in succeeding rules. 

|i=^ (170.) 

Rule XXXIX. — Preliminary. When the weight is eqically 
distributed. One-half of the quotient obtained by the preced- 
ing rule is a quantity which may be represented by K^ referred 
to in succeeding rules. 

Rule XL. — Preliminary. When the weight is per foot su- 
perficial. Multiply the weight per foot superficial, in pounds, 
by the square of the length, in feet, and by the distance apart 
from centres between two parallel beams, and divide the pro- 
duct by twice the value of aS, Table YL, multiplied by a deci- 
mal in proportion to unity as the safe weight is to the break- 
ing weight, and the quotient is a quantity which may be re- 
presented by Z, referred to in succeeding rules. 

Rule XLI. — Preliminary. When the weight is concentrated 
at any point in the length. Multiply the distance, in feet, 
from the loaded point to one support, by the distance, in feet, 
from the same point to the other support, and by four times 
the weight in pounds, and divide the product by the value of 
8^ Table YL, multiplied by a decimal in proportion to unity 
as the safe weight is to the breaking weight, and by the length, 



FRAMING. 231 

in feet ; and the quotient is a quantity wliich may be repre* 
sented by Q^ referred to in the rules. 

^^=Q (m.) 

Rule XLII.^ — Preliminary. When two or more weights are 
concentrated at any points in the length of the heams. Multi- 
ply each weight by each of the two parts, in feet, into which 
the length is divided by the point at which the weight is 
located, and divide four times the sum of these products by 
the value of S^ Table YL, multiplied by a decimal in propor- 
tion to unity as the safe weight is to the breaking weight, and 
by the length, in feet, and the quotient is a quantity which 
may be represented by F", referred to in the rules. 
4.{Rmn + Pgh-^ c&c.) ^ y .^ 

Sal ^ '^ 

Rule XLITI. — Preliminary. When the learn is not laid 
horizontal^ hut inclining. In the five preceding preliminary 
rules, multiply the result there obtained by the horizontal dis- 
tance between the supports, in feet, and divide the product by 
the length, in feet, and the quotient in each case is to be used 
for beams when inclined, as referred to in succeeding rules. 



TO FIND THE DIMENSIONS. 

Ride XLIY. — When the beam is rectangular. To find the 
depth. Divide the quantity represented in preceding rules by 
t/", K^ Z, Q^ or F", by the breadth, in inches, and the square 
root of the quotient will be the depth required in inches. 

To find the Ireadth. Divide the quantity represented by J", 
K^ Z, Q, or F, by the square of the depth, and the quotient 
will be the required breadth, in inches. 

To find both Ireadth and depths when they are to be in a 
given proportion. Divide the quantity represented by J, K^ 
Z, §, or F, by a decimal in proportion to unity as the breadth 



232 AMERICAN HOUSE-CARPENTER. 

]s to be to tlie depth, and the cube root of the quotient Avill be 
the depth in inches. Multiply the depth by the aforesaid de- 
cimal and the quotient will be the hreadtf. in inches. 

Example. — A locust beam, 10 feet long in the clear of the 
supports, is required to sustain safely 3,000 pounds at the 
middle of its length, the portion of the breaking weight allow- 
able being 0-3 ; what is the required breadth and depth ? 
Proceeding by the rule for weight concentrated at middle, 
(Rule XXXYIII.,) 3,000, the weight, by 10, the length, equals 
30,000. The value of S, Table YL, for locust, is 712 : this by 
0-3 the decimal, as above, equals 222*6 ; the 30,000 aforesaid 
divided by this 222-6 equals 131*77, equals the quantity repre- 
sented by J. Now to find the depth when the breadth is 4 
inches, 131*77 divided b}^ 4, the breadth, as above required, 
the quotient is 33*69, and the square root of this, 5*8, is the 
required depth in inches. But to find the breadth, when the 
depth is known, let the depth be 6 inches, then 131*77 
divided by 36, the square of the depth, equals 3*745 the breadth 
required in inches. Again, to find both breadth and depth in 
a given proportion, say, as 0*6 is to I'O. Here 131*77 divided 
by 0*6 equals 221*617, the cube root of which is 6*08, the re- 
quired depth in inches, and 6*08 by 0*6 equals 3*618, the re- 
quired breadth in inches. 

Thus it is'seen, in this example, that a piece of locust tim- 
ber, 10 feet long, having 3,000 pounds conGsntrated at the 
middle of its length, as fV of its breaking load, is required to 
be 1 by 5ff inches, or 3f by 6 inches, or 3| by 6-g- inches. If 
this load were equally diffused over the length, the dimensions 
required would be found to be 4 by 4*1, or 1*87 by 6, or 2*895 
Dy 4*825 inches, in the three cases respectively. 

Example. — A tier of chestnut beams, 20 feet long, placed 
cne foot apart from centres, is required to sustain 100 pounds 
per superficial foot upon the floor laid upon them : this load 
to be 0*2 of the breaking weight ; what is the required dimen- 



FRAMING. 233 

sions of the cross-section ? By Kule XL., the rule for a load 
per foot superficial, 100 by 20 x 20 and by 1 equals 40,000. 
Twice 503, the value of jS for chestnut, Table YI., by 0*2 
equals 201-2. The above 40,000 divided by 201-2 equals 198*8, 
the value of Z. l!^ow if the breadth is known, and is 3 inches, 
198*8 divided by 3 equals 66"27, the square root of which ia 
8-14:, the required depth. But if the depth is known and is 9 
inches, 198-8 divided by (9x9=:) 81 equals 2*154: inches, the 
required breadth. Again, when the breadth and depth are 
required in the proportion of 0-25 to I'O, then 198*8 divided 
by 0*25 equals 795*2, the cube root of which is 9*265, the 
required depth in inches, and 9-265 by 0-25 equals 2*316, the 
required breadth in inches. 

Exatnple. — A cast-iron bar, 10 feet long, is required to sus- 
tain safely 5,000 pounds placed at 3 feet from one end, and 
consequently at 7 feet from the other end, the portion of the 
breaking load allowable being 0*3 ; what must be the size of 
the cross section ? By Rule XLL, the rule for a concentrated 
load at any point in the length of the beam, 3 x 7 X 4 x 5000 
= 420,000. And 1926, the value of S for cast-iron. Table YL, 
by 0-3 and by 10 equals 5778. The aforesaid 420,000 divided 
by this, 5778, equals 72-689, the value of Q. l^ow if the 
breadth is fixed at 1*5 then 72*689 divided by 1*5 equals 
48-459, the square root of which is 6-96, the required depth in 
inches. But if the depth is fixed at 6 inches, then 72-689, the 
value of Q^ divided by 36, the square of 6, equals 2-019, the 
required breadth in inches. Again, if the breadth and depth 
are required in the proportion 0-2 to I'O ; then Q^ 72-689, 
divided by 0-2, equals 363-445, the cube root of which is 7-136, 
the required depth in inches ; and 7*136 by 0*2 equals 1*427 
the required breadth in inches. 

Rule XLY. — When the beam is square to find the breadth 
of a side. The cube root of the quantity represented by «/, K, 
Z, Q^ or F, in preceding rules, is the breadth of the side required 

30 



234: AMERICAN HOUSE- CAEPENTER. 

Exairvple. — A Georgia pine beam, 10 feet long, is required 
to sustain, as 0'3 of the breaking load, a weight of 30,000 
pounds equally distributed over its length, and the beam to be 
square, what must be the breadth of the side of such a beam ? 
By the rule for an equally distributed load, (Rule XXXIX.,) 
30,000 X 10 == 300,000, and 510 (the value of S, for Georgia 
pine, Table YL) x 0-3 = 153. 300,000 divided by 153 equals 
1960-781, and one-half of this equals 980-392, the value of ^ 
Now the cube root of this is 9-931 inches, or 9]-f, the required 
side. Had the weight been concentrated at the middle 
1960*781 would be the value of «/, and the cube root of this 
12*515, or 12-| inches, would be the size of a side of the 
beam. 

Example. — A square oak beam, 20 feet long, is required to 
sustain, as 0-25 of the breaking strength, three loads, one of 
8,000 pounds at 5 feet from one end, one of 7,000 pounds at 
14 feet, and one of 5,000 pounds at 8 feet from one end, what 
must be the breadth of a side of the beam? The value oi 8^ 
for oak, Table YL, is 571. Ey the rule for this case, (Rule 
XLn.,) 8000 X 5 X 15 equals 600,000 ; and 7000 X 11 x 6 
equals 588,000 ; and 5000 x 8 x 12 equals 180,000. The sum 
of these products is 1,668,000 ; this by 4 equals 6,672,000. 
Now 571 X 0-25 X 20 equals 2870, and the 6,672,000 divided 
by the 2870 equals 2325, the number represented by V ; the 
cube root of which is 13-25, the required size of a side of the 
beam, 13^ inches. This is for a horizontal beam. Now if 
this beam be laid inclining, so that the horizontal distance 
between the bearings is 15 feet, then to find the size by the 
rule for this case, viz. XLIII., the above number "F, equal to 
2325, multiplied by 15, the horizontal distance, equals 31,875, 
and this divided by 20, the length, equals 1743-75. Now by 
Rule XLY., the cube root of this is 12*04, the required size of 
a side — 12 inches full. 

Rule XLYI. — When the beam is round. Divide the quan- 



FRAMING. 235 

tity represented by J^ K^ Z, Q^ or Fby tlie decimal 0'589, and 
the cube root of the quotient will be the required diameter. 

Example. — A white pine beam or pole, 10 feet long, is re- 
quired to sustain, as the 0*2 of the breaking strength, a load 
of 5,000 pounds concentrated at the middle, what must be the 
diameter ? The value of B^ for white pine, Table YL, is 390. 
Now by the rule for load at middle, (XXXYIII.,) 5000 x 10 
= 50,000 ; and 390 x 0-2 = 78 ; and 50,000 -^ 78 == 641 = J, 
By this rule, 611 -f- 0-589 = 1088*28, the cube root of which, 
10"28, is the required diameter. If this beam be inclined, 
so that the horizontal distance between the supports is 7 feet, 
then to find the diameter, by Rule XLIII., value of J as 
above, 611 multiphed by 7 and divided by 10 equals 4:4:8'7. 
ISTow by this rule, 418-7 ^ 0*589 =: 761*796, the cube root of 
which, 9*133, is the required diameter. 

Example. — A spruce pole, 10 feet long, is required to sus- 
tain, as the 0*33^ or \ of the breaking weight, a load of 1,000 
pounds at 3 feet from one end, what must be the diameter % 
The value of B for spruce, Table ^^.^ is 315. By the rule for 
this case, (Rule XLL,) 3 x 7 x 4 x 1000 = 84,000 ; and 345 
X i X 10 r:r 1150 ; and 84,000 -^ 1150 =. 93*04, the value of Q, 
Now by this rule, 73*04 ^ '589 = 124, the cube root of which, 
4*9866, is the required diameter in inches. 

335. — Systems of FroMiing. In the various parts of framing 
known as floors, partitions, roofs, bridges, &c., each has a spe- 
cific object ; and, in all designs for such constructions, this 
object should be kept clearly in view ; the various parts being 
so disposed as to serve the design with the least quantity of 
material. The simplest form is the best, not only because it is 
the most economical, but for many other reasons. The great 
number of joints, in a complex design, render the construction 
liable to derangement by multiplied compressions, shrinkage, 
and, in consequence, highly increased oblique strains ; by 
which its stability and durability are greatly lessened. 



^36 



AMEBIC Aiq HOUSE-CAEPENTEK. 



FLOOES. 



336. — Floors at'e most generally constructed single, that is, 
gimplj a series of parallel beams, each spanning the width oi 




Fig 224. 



the floor, as seen at Fig, 224. Occasionally floors are con 




Fig. 225. 



structed double, as at Fig. 225 ; and sometimes framed, as af 



FRAMING. 



237 




Fig. 226 ; but these methods are seldom practised, inasmucli 
as either of these require more timber than the single floor. 
Where lathing and plastering is attached to the floor beams to 
form, a ceiling below, the springing of the beams, by custo- 
mary use, is liable to crack the plastering. To obviate this in 
good dwellings, the double and framed floors have been 
resorted to, but more in former times than now, as the cross- 
furring (a series of narrow strips of board or plank, nailed 
transversely to the underside of the beams to receive the lath- 
ing for the plastering,) serves a like purpose very nearly as well. 
337. — In single floors the dimensions of the beams are to be 
ascertained by the preceding rules for the stiffness of materials- 
These rules give the required dimensions for the various kinds 
of material in common use. The rules may be somewhat 
abridged for ordinary use, if some of the quantities repre- 
sented in the formula be made constant within certain limits. 
For example, if the load per foot superficial, and the rate of 
deflection, be flxed, then these, together with the f , and the 



238 AMERICAN HOUSE-CAKPENTER. 

constant represented hj II] may be reduced to one constant. 
For dwellings, tlie load per foot may be taken at 66 pounds, 
as this is the weight, that has been ascertained by experiment, 
to arise from a crowd of people on their feet. To this add 20 
for the weight of the material of which the floor is composed, 
and the sum, 86, is the value of /*, or the weight per foot 
superficial for dwellings. The rate of deflection allowable for 
this load may be flxed at 0*03 inch per foot of the length. 
Then (44) transposed, 

6fGl 



73 

S An 



becomes 



5 X 86 ^ _T^ 
8 X 0-03 X _^ - ^ ^ 



which, reduced, is 

l^XGr = 'bd' (175.) 

1800 * 

Eedncing — ^ for five of the most common woods, and there 

results, rejecting small decimals, and putting —p~ = x, x 
equal, for 

Georgia pine 0*6 

Oak . . O'l 

White pine 1*0 

Spruce 1'15 

Hemlock 1*45 

Therefore, the rule is reduced to x c T = h d^. And for 
white jpine^ the wood most used for floor beams, a? = 1*0, and 
therefore disappears from the formula, rendering it still more 
simple, thus, 

cr=.h d' (176.) 

The dimensions of beams for stores, for all ordinary business, 
may also be calculated by this modified rule, (175,) for it will 
require about 3^ times the weight used in this rule, or about 



FRAMING. 239 

300 pounds, to increase the deflection to the limit of elasticity 
in white pine, and nearly that in the other woods. But for 
warehouses, taking the rate of deflection at its limit, and fixing 
the weight per foot at 500 pounds, including the weight of the 
material of which the floor is constructed, and letting y repre- 
sent the constant, then 

ycV^ld' (177.) 

and y equals, for 

Georgia pine 1-35 

Oak .......... 1-35 

White pine 1-V5 

Spruce 2*2 

Hemlock 2*85 

338. — Hence to find the dimensions of floor beams for dwellr 
ings when the rate of deflection is 0*03 inch per foot, or for 
ordinary stores when the load is about 300 pounds per foot, 
and the deflection caused by this weight is within the limits 
of the elasticity of the material, we have the following rule :, 

Bule XLYII. — Multiply the cube of the length by the dis- 
tance apart between the beams, (from centres,) both in feet, 
and multiply the product by the value of a?, {Art. 337.) l^ow 
to find the breadth, divide this product by the cube of the 
depth in inches, and the quotient w^ill be the breadth in inches. 
But if the depth is sought, divide the said product by the 
breadth in inches, and the cube root of the quotient will be 
the depth in inches ; or, if the breadth and depth are to be in 
proportion, as r is to unity, r representing any required deci- 
mal, then divide the aforesaid product by the value of r, and^otf^i 
extract the square root of the quotient, and the square root of 
this square root will be the depth required in inches, and the 
depth multiplied by the value of r will be the breadth in 
inches. 

Example. — To find the 'breadth. In a dwelling or ordinary 
store what must be the breadth of the beams, when placed 15 



J fl < / 



240 AMERICAN HOUSE-CAEPENTEE. 

inches from centres, to support a floor covering a span of IG 
feet, the depth being 11 inches, the beams of oak ? Bj the 
rule, 4096, the cube of the length, by li, the distance from 
centres, and by 0*7, the value of a?, for oak, equals 3584. This 
divided by 1331, the cube of the depth, equals 2*69 inches, or 
2|i inches, the required breadth. 

Example. — To find the depth. The conditions being the 
same as in the last example, what miist be the depth when the 
breadth is 3 inches. The product, 3584, as above, divided by 
3, the breadth, equals 1194f ; the cube root of this is 10*61, or 
lOf inches nearly. 

Example. — To find the hreadth and depth in proportion^ 
say, as 0-3 to I'O. The aforesaid product, 3584, divided by 
0"3, the value of r, equals ll,946f , the square root of which is 
109*3, and the square root of this is 10*45, the required depth. 
This multiplied by 0*3, the value of ^, equals 3*135, the re- 
quired breadth, the beam is therefore to be 3-|- by 10^ inches. 

339. — And to find the breadth and depth of the beams for a 
floor of a warehouse sufficient to sustain 500 pounds per foot 
superficial, (including weight of the material in the floor,) with 
a deflection not exceeding the limits of the elasticity of the 
material, we have the following rule : 

Bute XLYIII. — ^The same as XLYIL, with the exception 
that the value of y {Art. 337) is to be used instead of the 
value of X. 

Example. — Tofim,d the "breadth. The beams of a warehouse 
floor are to be of Georgia pine, with a clear bearing between 
the walls of 15 feet, and placed 14 inches from centres, what 
must be the breadth when the depth is 11 inches ? By the 
rule, 3375, the cube of the length, and 1^, the distance from 
centres, and 1*35, the value of y^ for Georgia pine, all multi- 
plied together, equals 6315-625 ; and this product divided by 
1331, the cube of the depth, equals 3*994, the required depth, 
or 4 inches. 



FEAMrNG. 241 

Examjple. — To Jmd the d^jyth. The conditions remaining, 
as in last example, what must be the depth when the breadth 
is 3 inches? 5315*625, the said product, divided by 3, the 
breadth, equals lYYl-875, and the cube root of this, 1-2-1, or 12 
inches, is the depth required. 

Example. — To find the breadth and depth in a given projpor- 
tion, saj, 0-35 to I'O. 5315-625 aforesaid, divided by 0*35, the 
value of 7', equals 15187*5, the square root of which is 121*8, 
and the square root of this square root is 11*04, or 11 inches, 
the required depth. And 11*04 multiplied by 0*35, the value 
of r, equals 3*864, the required breadth — 3-g- inches. 

340. — It is sometimes desirable, when the breadth and depth 
of the beams are fixed, or when the beams have been sawed 
and are now ready for use, to know the distance from centres 
at which such beams should be placed, in order that the floor 
be sufficiently stiff. In this case, (1Y5,) transposed, and put- 
ting X — — ^, there results 

= ^' (178.) 

xl ^ ^ 

This in words, at length, is, as follows : 

Bule XLIX. — Multiply the cube of the depth by tho 
breadth, both in inches, and divide the product by the cube 
of the length, in feet, multiplied by the value of a?, for dwell- 
ings, and for ordinary stores, or by y for warehouses ; and the 
quotient will be the distance' apart from centres in feet. 

Example. — A span of 17 feet, in a dwelling, is to be covered 
by white pine beams, 3 X 12 inches, at what distance apart 
from centres must they be placed ? By the rule, 172S, the 
cube of the depth, multiplied by 3, the breadth, equals 5184. 
The cube of 17 is 4913, this by 1*0, the value of cc, lor white 
pine, equals 4913. The aforesaid 5184, divided by this, 4913, 
equals 1*055 feet, or 1 foot and | of an inch. 

341. — Where chimneys, fl'jes, stairs, etc., occur to interrupt 

^1 



242 



AMERICAN HOUSE-CAKPENTER. 




the bearing, tlie beams are framed into a piece, 5, {Fig. 227,) 
called a header. The beams, a «, into- which the header is 
framed, are called t/rimmers or carriage-'beams. These framed 
beams require to be made thicker than the common beams. 
The header must be strong enough to sustain one-half of the 
weight that is sustained uvon the tail beams, c <?, (the wall at 
the opposite end or another header there sustaining the other 
half,) and the trimmers must each sustain one-half of the 
weight sustained by the header in addition to the weight it 
supports as a common beam. It is usual in practice to make 
these framed beams one inch thicker than the common beams 
for dwellings, and two inches thicker for heavy stores. This 
practice in ordinary cases answers very well, but in extreme 
cases these dimensions are not proper. Eules applicable gene- 
rally must be deduced from the conditions of the case — the 
load to be sustained and the strength of the material. 

342. — For the header, formula (68,) Table Y., is applicable. 
The weight, represented by w^ is equal to the superficial area 
of the floor supported by the header, multiplied by the load 
on every superficial foot of the floor. This is equal to the 
length of the header multiplied by half the length of the tail 
beams, and by 86 pounds for dwellings and ordinary stores, or 



FKAMING. 243 

by 500 pounds for warehouses. Calling tlie length of the taH 
beams, in feet, ^, formula (68,) becomes 

Then if/ equals 86, and n equals 0*03, there results 
1 = '-^ (179.) 

This in words, is, as follows : 

Rule L. — Multiply 900 times the length of th6 tail beams 
by the cube of the length of the header, both in feet. The 
product, divided by the cube of the depth, multiplied by the 
value of E^ Table IL, will equal the breadth, in inches, for 
dwellings or ordinary stores. 

Exam,jple. — A header of white pine, for a dwelling, is 10 feet 
long, and sustains tail beams 20 feet long, its depth is 12 
inches, what must be its breadth ? By the rule, 900 X 20 x 
10' = 18,000,000. This, divided by (12" X 1Y50 =) 3,024,000, 
equals 5*95, say 6 inches, the required breadth. 

For heavy warehouses the rule is the same as the above, 
only using 1550 in the place of the 900. This constant may 
be varied, at discretion, to anything between 900 and 5000, in 
accordance with the use to which the floor is to be put. 

343. — In regard to the trimmer or carriage beam, formula 

(136,) Table YIII., is applicable. The load thrown upon the 

trimmer, in addition to its load as a common beam, is equa 

to one-half of the load on the header, and therefore, as has 

been seen in last article, is equal to one-half of the superlicial 

area of the floor, supported by the tail beams, multiplied by 

the weight per superficial foot of the load upon the floor; 

therefore, when the length of the header, in feet, is represented 

'in 
hjj, and the length of the tail beam by n, w equals 'L-^ -~xf 

2 2 

equals ^fj n, and therefore (136,) of Table YIII., becomes 
^-Sad'l 



24:4 AMERICAN HOrSE-CAEPENTER. 

equals the additional tliickness to be given to a common 
beam wlien used as a trimmer, and for dwellings when f 
equals 86 and a equals 0*3, this part of the formula reduces to 
286 J, or, for simplicity, call it 300, which would be the same 
as fixing/ at 90 instead of 86. Then we have 

^ = -^dnr ^^^^'^ 

This, in words, is as follows : 

Bule LI. — For dwellings. Multiply 300 times the length 
of the header by the square of the length of the tail beams, 
and by the difference in length of the trimmer and tail beams, 
all in feet. Divide this product by the square of the depth in 
inches, multiplied by the length of the trimmer in feet, and by 
the value of S^ Table YL, and the quotient added to the thick- 
ness of a common beam of the floor, will equal the required 
thickness of the trimmer beam. 

Example. — A tier of 3 x 12 inch beams of white pine, hav- 
ing a clear bearing of 20 feet, has a framed well-hole at one 
side, of 5 by 12 feet, the header being 12 feet long, what must 
be the thickness of the trimmer beams ? By the rule, 300 x 
12 X 15' X 5, divided by the product of 12' x 20 x 390, equals 
3*6, and this added to 3, the thickness of one of the common 
beams, equals %'Q^ the breadth required, 6-|- inches. 

For stores and warehouses the rule is the same as the above, 
only the constant, 300, must be enlarged in proportion to the 
load intended for the floor, making it as high as 1600 for 
heavy warehouses. 

341. — When a framed opening occurs at any point removed 
from the wall, requiring two headers, then the load from the 
headers rest at two points on the carriage beam, and here for- 
mula (141.) Table YIIL, is applicable. In this special case 
this formula reduces to 

._ ZQiOj{m'n+h'g) . 



FRA^IING. 24:5 

vvliere h equals the additicnial thickness, in inches, to be given 
to the carriage beam over the thickness of the common beams ; 
y, the length of the header, in feet ; m and h the length, in 
feet, respectively, of the two sets of tail beams, and m^ n = 'k 
+ g = l. 

Tlie constant in the above, (181,) is for dwellings ; if the 
floor is to be loaded more than dwelling floors, then it must be 
increased in proportion to the increase of load up to as high as 
1600 for warehmcses. 

Rule LIl. — Trimmer teams for framed openings occurring 
so as to require two headers. Multiply the square of the 
length of each tail beam by the difl'erence of length of the tail 
beam and trimmer, all in feet, and add the products ; multiply 
their sum by 300 times the length, in feet, of the lieader, and 
divide this product by the product of the square of the depth, 
in inches, by the length, in feet, and by the value of S, Table 
yi. ; and the quotient, added to the thickness of a common 
beam of the tier, will equal the thickness of the trimmer 
beams. 

Example. — A tier of white pine beams, 4 X 14 inches, 20 
feet long, is to have an opening of 5 x 10 feet, framed so that 
the length of one series of tail beams is Y feet, the other 8 feet, 
what must be the breadth of the trimmers ? Here, (7*^ X 13) 
+ (8' X 12) equals 1405. This by 300 x 10 equals 4,215,000. 
This divided by 1,528,800 (== 14^^ + 20 X 390) equals 2-75, and 
this added to 4, the breadth, equals 6*75, or 6|-, the breadth 
required, in inches. 

345. — Additional stiffness is given to a floor by the insertion 
of hridging strips, or struts, as at a a^ {Fig. 228.) These pre- 
vent the turning or twisting of the beams, and w^hen a weight 
is placed upon the floor, concentrated over one beam, they 
prevent this beam from descending below the adjoining beams 
to the injury of the plastering upon the underside. It is usual 
to insert a course of bridging at every 5 to 8 feet of the length 



246 



AMERICAN HOUSE-CARPENTER. 




Fig. 228. 



c.f the beam. Strips of board or plank nailed to the underside 
of the floor beams to receive the lathing, are termed cross- 
furring, and should not be over 2 inches wide, and placed 12 
inches from centres. It is desirable that all furring be narrow, 
in order that the clinch of the mortar be interrupted but little. 
When it is desirable to prevent the passage of sound, the 
openings between the beams, at about 3 inches from the upper 
edge, are closed by short pieces of boards, which rest on elects, 
nailed to the beam along its whole length. This forms a floor, 
on which mortar is laid from 1 to 2 inches deep. This is 
called deafening. 

346. — When the distance between the walls of a building is 
great, it becomes requisite to introduce girders, as an addi- 
tional support, beneath the beams. The dimensions of girders 
m.ay be ascertained by the general rules for stiffness. For- 
mulas (72,) (73,) and (74,) Table Y., are applicable, taking y, 
at 86, for dwellings and ordinary stores, and increased in pro- 
portion to the load, up to 500, for heavy warehouses. When 
but one girder occurs, in the length of the beam, the distance 
from centres, c, is equal to one-half the length of the beam. 

347. — When the breadth of a girder is more than about 12 
inches, it is recommended to divide it by sawing from end to 
ead, vertically through the middle, and then to bolt it to 



FRAMING. 247 

gether witli the sawn sides outwards. This is not tb strength- 
en the girder, as some have supposed, but to reduce the size 
of the timber, in order that it may dry sooner. The opera- 
tion affords also an opportunity to examine the heart of the 
stick — a necessary precaution ; as large trees are frequently 
in a state of decay at the heart, although outwardly they are 
seemingly sound. When the halves are bolted together, thin 
slips of wood should be inserted between them at the several 
points at which they are bolted, in order to leave sufficient 
space for the air to circulate between. This tends to prevent 
decay ; w^iich will be found first at such parts as are not 
exactly tight, nor yet far enough apart to permit the escape 
of moisture. 

348. — When girders are required for a long bearing, it is 
usual to truss them ; that is, to insert between the halves two 
pieces of oak which are inclined towards each other, and 
which meet at the centre of the length of the gi^'der, like the 
rafters of a roof-truss, though nearly if not quite concealed 
within tlie girder. This, and many similar methods, though 
extensively practised, are generally worse than useless ; since 
it has been ascertained that, in nearly all such cases, the ope- 
ration has positively weakened the girder. 

A girder may bo strengthened by mechanical contrivance, 
when its depth is required to be greater than any one piece of 




Fig. 229. 



24:8 AMERICAN HOUSE-CARPENTER. 

timber will allow. Fig. 229 shows a very simple jet mvalu 
able method of doing this. The two pieces of which the gir« 
der is composed are bolted, or pinned together, having keys 
inserted between to prevent the pieces from sliding. The 
keys should be of hard wood, well seasoned. The two pieces 
should be about equal in depth, in order that the joint be- 
tween them may be in the neutral line. (See Art. 317.) The 
thickness of the keys should be about half their breadth, and 
the amount of their united thicknesses should be equal to a 
trifle over the depth and one-third of the depth of the girder. 
Instead of bolts or pins, iron hoops are sometimes used ; and 
when they can be procured, they are far preferable. In this 
case, the girder is diminished at the ends, and the hoops 
driven from each end towards the middle. 

34:9.- — Beams may be spliced, if none of a sufficient length 
can be obtained, though not at or near the middle, if it can 
be avoided. (See Art. 281.) Girders should rest from 9 to 
12 inches on the wall, and a space should be left for the air 
to circulate around the ends, that the dampness may evapo- 
rate. Eloor-timbers are supported at their ends by walls of 
considerable height. They should not be permitted to rest 
upon intervening partitions, which are not likely to settle as 
much as the walls ; otherwise the unequal settlements will 
derange the level of the floor. As all floors, however well- 
constructed, settle in some degree, it is advisable to frame 
the beams a little higher at the middle of the room. than at 
its sides, — as also the ceiling-joists and cross-furring, when 
either are used. In single floors, for the same reason, the 
rounded edge of the stick, if it have one, should be placed 
uppermost. 

If the floor-plank are laid down temporarily at first, and left 
to season a few months before they are finally driven together 
and secured, the joints will remain much closer. But if the 
edges of the plank are phined after the first laying, they will ' 



FKAMING. 249 

shrink again ; as it is the nature of wood to shrink after every 
planing however dry it may have been before. 



PARTITIONS. 

350. — Too Kttle attention has been given to the construction 
of this part of the frame-work of a house. The settling of 
floors and the cracking of ceilings and walls, which disfigure 
to so great an extent the apartments of even our most costly 
houses, may be attributed almost solely to this negligence, A 
square of partitioning weighs nearly a ton, a greater weight, 
when added to its customary load, such as furniture, storage, 
&c., than any ordinary floor is calculated to sustain. Hence 
the timbers bend, the ceilings and cornices crack, and the 
whole interior part of the house settles ; showing the necessity 
for providing adequate supports independent of the floor- 
timbers. A partition should, if practicable, be supported by 
the walls with which it is connected, in order, if the walls set- 
tle, that it may settle with them. This would prevent the 
separation of the plastering at the angles of rooms. For the 
same reason, a firm connection with the ceiling is an im- 
portant object in the construction of a partition. 

351. — The joists in a partition should be so placed as to dis- 
charge the weight upon the points of support. All oblique 
pieces in a partition, that tend not to this object, are much 
better omitted. Fig. 230 represents a partition having a door 
in the middle. Its construction is simple but e^fl'ective. Fig. 
231 shows the manner of constructing a partition having doors 
near the ends. The truss is formed above the door-heads, 
and the lower parts are suspended from it. The posts, a and 
5, are halved, and nailed to the tie, c d^ and the sill, e f. The 
braces in a trussed partition should be placed so as to form, as 
near as possible, an angle of 40 degrees with the horizon. In 
partitions that are intended to support only their own weight, 

32 



250 



AMERICAN HOUSE-CARPENTER. 




Fig. 230. 



ri 






1 


\m 


'f i 


^H^L 




1 ^ 


:• 


>; "^ 


1 




1 


:k_._._. . 




b 


a 


h 


L 



UiMi " 



Fig. 231. 



the principal timbers may be 3 X 4 inches for a 20 feet span, 
8-i X 5 for 30 feet, and 4 x 6 for 40. Tlie thickness of the 
filling-in stuff may be regulated according to what is said at 
Art. 345, in regard to the width of fm-ring for plastering. 
The filling-in pieces should be stiffened at about every three 
feet by short struts between. 

All superfluous timber, besides being an unnecessary load 
upon the points of support, tends to injure the stability of the 
plastering ; for, as the strength of the plastering depends, in a 
great measure, upon its clinch, formed by pressing the mortar 



FRAMING. 



251 



through the space between the laths, the narrower the surface, 
therefore, upon which the laths are nailed, the less will be the 
quantity of plastering unclinched, and hence its greater secu- 
rity from fractures. For this reason, the principal timbers of 
the partition should have their edges reduced, by chamfering 
off the corners. 



V- 



^; 



^ 



^ 



~-i- 



^ 



\ 



^j 



Fig. 232. 

352. — "When the principal timbers of a partition require to 
be large for the purpose of greater strength, it is a good plan 
to omit the upright filling-in pieces, and in their stead, to 
place a few horizontal pieces ; in order, upon these and the 
principal timbers, to nail upright battens at the proper dis- 
tances for lathing, as in Fig. 232. A partition thus con- 
structed requires a little more space than others ; but it has 
the advantage of insuring greater stability to the plastering, 
and also of preventing to a good degree the conversation of 
one room from being heard in the other. When a partition is 
required to support, in addition to its own weight, that of a 
floor or some other burden resting upon it, the diraer'^ons of 
the timbers may be ascertained, by applying the principles 
which regulate the laws of pressure and those of the resistance 
of timber, as explained at the first part of this section. The 
following data, however, may assist in calculating the amount 
of pressure upon partitions : 



252 AMERICAN HOrSE-CAEPEXTEK. 

White pine timber weighs from 22 to 32 pounds per cubic 
foot, varying in accordance with tlie amount of seasoning it 
has had. Assuming it to weigh 30 pounds, the weight of the 
beams and floor plank in every superficial foot of the flooring 
will be, when the beams are 

3x8 inclies, and placed 20 inclies from centres, 6 pounds. 
3 X 10 " " " 18 " " " 7i " 

3 X 12 " " " 16 " " " 9 " 

3 X 12 " " " 12 " " " 11 ** 

4 X 12 " «* " 12 " " " 13 
4 X 14 " " " 14 " " " 13 

In addition to the beams and plank, there is generally the 
plastering of the ceiling of the apartments beneath, and some- 
times the deafening. Plastering may be assumed to weigh 9 
pounds per superficial foot, and deafening 11 pounds. 

Hemlock weighs about the same as white ^ine. A parti- 
tion of 3 X 4 joists of hemlock, set 12 inches from centres, 
therefore, will weigh about 2-J- pounds per foot superficial, and 
when plastered on both sides, 20J pounds. 

353. — AVhen floor beams are supported at the extremities, 
and by a partition or girder at any point between the extre- 
mities, one-half of the weight of the whole floor will then be 
supported by the partition or girder. As the settling of parti- 
tions and floors, which is so disastrous to plastering, is fre- 
quently owing to the shrinking of the timber and to ill-made 
joints, it is very important that the timber be seasoned and 
the work well executed. Where practicable, the joists of a 
partition ought to extend down between the floor beams to 
the plate of the partition beneath, to avoid the settlement con- 
sequent upon the shrinkage of the floor beams. 

KOOFS.* 

354:. — In ancient biildings, the Korman and the Gothic, the 

• See also AH. 238. 



TEAMING. 



253 



walls and buttresses were erected so massive and firm, that it 
was customary to construct their roofs without a tie-beam: 
the walls being abundantly capable of resisting the lateral 
pressure exerted by the rafters. But in modern buildings, the 
walls are so slightly built as to be incapable of resisting 
scarcely any oblique pressure ; and hence the necessity of 
constructing the roof so that all oblique and lateral strains 
may be removed; as, also, that instead of having a tendency 
to separate the walls, the roof may contribute to bind and 
steady them. 

355. — In estimating the pressures upon any certain roof, for 
the purpose of ascertaining the proper sizes for the timbers, 
calculation must be made for the pressure exerted by the wind, 
and, if in a cold climate, for the weight of snow, in addition to 
the weight of the materials of which the roof is composed., 
The weight of snow will be of course according to the depth 
it acquires. Snow weighs 8^1bs. per cubic foot, and more 
when saturated with water. In a severe climate, roofs ought 
to be constructed steeper than in a milder one, in order that 
the snow may have a tendency to slide off before it becomes 
of sufficient weight to endanger the safety of the roof. The 
inclination should be regulated in accordance with the qualities 
of the material with which the roof is to be covered. Tlie 
following table may be useful in determining the smallest in- 
clination, and in estimating the weight of the various kinda 



ot covering : 



Material. 


InoHnntion. 


Weight upon a 
square foot. 


Tin 

E:,T". ::;:;: 

Zinc 

Short pine shingles .... 
Long cypress shingles .... 
Slate 


Rise 1 inch to a foot 
" 1 " " " 
" 2 inches " " 
" 3 " " " 
u 5 " « <« 

" 6 " " " 
{( g « (( « 


f to l^lbs. 

1 to H " 

4 to 7 " 
H to 2 " 
H to 2 " 

2 to 3 « 

5 to 9 " 



^54 AMERICAN HO •;SE-CARPENTER. 

The weight of the covering, as above estimated, is that of 
the material only, with the weight of whatever is used to fix 
it to tlie roof, such as nails, &c. What the material is laid 
on, such as plank, boards or lath, is not included. The weight 
of plank is about 3 pounds per foot superficial ; of boards, 2 
pounds ; and lath, about a half pound. 

356. — The weights and pressures on a roof arise from the 
roofing, the truss, the ceiling, wind and snow, and may be 
stated as follows : 

First^ the Roofing, — On each foot superficial of the inclined 
surface, 

Slating -will -weigli about 7 lbs. 

Roof plank, 1^ inches thick . . « « <« 2-7 " 

Roof beams or jack rafters . . « « « 2*3 " 

Total, 12 lbs. 

'This is the weight per foot on the inclined surface ; but it is 
desirable to know how much per foot, measured horizontally^ 
this is equal to. The horizontal measure of one foot of the 
inclined surface is equal to the cosine of the angle of inclina- 
tion. Therefore, 

cos. \ \ \\ p \ w — -^ ; 

■^ COS. 

where jp represents the pressure on a foot of the inclined sur- 
face, and w the weight of the roof per foot, measured horizon- 
tally. The cosine of an angle is equal to the base of the right- 
angled triangle divided by the hypothenuse, which in this 
case would be half the span divided by the length of the 

rafter, or — -, where s is the span, and I the length -f the 

2 1 

rafter. Hence, 

P _ i? _ 2 Zjf? 



COS. 



6 



2' 

or, twice the pressure per foot of inclined surface, multiplied 



FRAMING. 256 

by the length of the rafter, and divided by the span, will give 
the weight per foot measured h'^rizontally ; or, 

24:1 =w (182.) 

s 

equals the weight per foot, measured horizontally, of the roof 

beams, plank, and covering for a slate roof 

Second, the Truss. — ^The weight of the framed truss is nearly 

in proportion to the length of the truss, and to the distance 

apart at which the trusses are placed. 

w = 5-2cs (183.) 

equals the weight, in pounds, of a white pine truss with iron 

suspension rods and a horizontal tie beam, near enough foi 

the requirements of our present purpose ; where s equals the 

length or span of the truss, and c the distance apart at which 

the trusses are placed, both in feet. It is desirable to know 

how much this is equal to per foot of the area over which the 

truss is to sustain a covering. This is found by dividing the 

weight of the truss by the span, and by the distance auart 

from centres at which the trusses are placed ; or, 

5:^^ = 6'2 = w (184.) 

cs ^ 

equals the weight in pounds per foot to be allowed for thej 
truss. 

Thi/rd, the Ceiling. — The weight supported by the tie beams, 
viz. : that of the ceiling beams, furring and plastering, is about 
9 pounds per superficial foot. 

Fourth, the Wind. — ^The force of wind has been known as 
high as 60 pounds per superficial foot against a vertical siir- 
face. The effect of a horizontal force on an inclined surface 
is in proportion to the sine of the angle of inclination, the ef- 
fect produced being in the direction at right angles to the in- 
clined surface. The force thus acting may be resolved into 
forces acting in two directions — the one horizontal, the other 
vertical; the former tending, in the case of a. roof, to thrust 



256 AMEEICAK HOUSE-CAKPENTEE. 

aside the walls on which the roof rests, and the latter acting 
directly on the materials of which the roof is constructed— 
this latter force being in proportion to the sine of the angle 
of inclination multiplied by the cosine. This is the vertical 
effect of the wind npon a roof, without regard to the surface 
it acts upon. The wind, acting horizontally through one foot 
superficial of vertical section, acts on an area of inclined sur- 
face equal to the reciprocal of the sine of inclination, and the 
horizontal measurement of this inclined surface is equal to the 
cosine of the angle of inclination divided by the sine. This is 
the horizontal measurement of the inclined surface, and the 
vertical force acting on this surface is, as above stated, in pro- 
portion to the sine multiplied by the cosine. Combining these, 
it is found that the vertical power of the wind is in proportion 
to the square of the sine of the angle of inclination. There- 
fore, if the power of wind against a vertical surface be taken 
at 50 pounds per superficial foot, then the vertical effect on a 
roof is equal to 

w) = 50 sin.'^ = 50 ^ (185.) 

for each piece of the inclined surface, the horizontal measure- 
ment of which equals one foot ; where I equals the length of 
the rafter, and h the height of the roof. 

Fifths Snoio.—-lL\\Q weight of snow will be in proportion to 
the depth it acquires, and this will be in proportion to the 
rigour of the climate of the place at which the building is to 
be erected. Upon roofs of most of the usual inclinations, 
snow, if deposited in the absence of wind, will not slide off. 
When it has acquired some depth, and not till then, it will 
have a tendency, in proportion to the angle of inclination, to 
slide off in a body. The weight of snow may be taken, there- 
fore, at its weight per cubic foot, 8 pounds, multiplied by the 
depth it is usual for it to acquire. This, in the latitude of ISTew 
York, may be stated at about 2^ feet. Its weight would, 



FRAMING. 257 

therefore, be 20 pounds per foot superficial, measured horizon- 
tally. 

357. — ^There is one other cause of strain upon a roof; namel}^ 
the load that may be deposited m the roof when used as a room 
for storage, or for dormitories. But this seldom occurs. When 
a case of this kind does occur, allowance is to be made for it 
as shown in the article on floors. But in the general rule, now 
under consideration, it may be omitted. 

358. — The following, therefore, comprehends all the pressures 
or weights that occur on roofs generally, per foot superficial ; 

I 
For roof beams, plank, and slate (182) . . . . 24 - lbs. 

" the truss (184) . 5*2 « 

** ceiling 9 " 

« wind (185) ^^ ^a " 

" snow, latitude of New York 20 " 

Having found the weight per foot, the total weight for any 
part of the roof is found by multiplying the weight per foot by 
the area of that part. This process will give the weight sup- 
ported by braces and suspension rods, and also that supported 
by the rafters and tie beam. But in these last two, only Tialf 
of the pressure of the vnnd is to be taken, for the wind will 
act only on one side of the roof at the same time. 

The vertical pressure on the head of a brace, then, equals 

TT = 4 c ^(6 - + 8-55 + 12-5 ^) (186.) 

7 A* 

And W z=.G jpn^ where j? equals 4^6- -|- 8*55 -f 12*5 ~ i, 

equals the weight per foot. 

And the aggregate load of the roof on each truss equals 

Tr = 4:C5(6- + 8-55-f6-25^) (187.) 

7 ^2 

And W =cc[s^ where c[ = 4^6 - + 8*55 -f- 6-25 — \ equals th« 

33 



258 AMERICAN HOUSE-CAHPENTEE. 

weight per foot ; where c equals the distance apart from centres 
at which the trusses are placed ; n the distance horizontally 
between the heads of the braces, or, if these are not located at 
equal distances, then n is the distance horizontally from a point 
half-way to the next brace on one side to a point half-way to 
the next brace on the other side ; I the length of the rafter ; s 
the span, and h the height — all in feet. 

359. — By the parallelogram of forces, the weight of the roof 
is in proportion to the oblique thrust or pressure in the axis of 
the rafter, as twice the height of the roof is to the length of the 
rafter; or, 

TT : ^ :: 2A : Z, or 

U : I ::W : B = ^, (188.) 

where R equals the pressure in the axis of the rafter. And 
the weight of the roof is in proportion to the horizontal thrust 
in the tie beam, as twice the height of the roof is to half the 
span; or, 

TT : ^:: 2A :i-, or 

2A:|::Tr:^=f|, (189.) 

where H equals the horizontal thrust in the tie beam; the 
value of W in (188) and (189) being shown at (187), and (187) 
being compounded as explained in Art. 356. The weight 
is that for a slate roof If other material is used for covering, 
or should there be other conditions modifying the weight in 
any particular case, an examination of Art. 356 will show how 
to modify the formula accordingly. 

360. — ^The pressures may be obtained geometrically, as 
shown in Fig. 233, where A B represents the axis of the tie 
beam, A C tha axis of the rafter, D E and FB the axes of the 
braces, and D G, FE^ and C B^ the axes of the suspension rods. 
In this design for a truss, the distance A B \& divided into three 



equal parts, and the rods located at the two points of division, 
G and E. Bj this arrangement the rafter AC\^ supported at 
equi-distant points, D and F, The point D supports the rafter 
for a distance extending half-way to A and half-way to F^ and 
the point F sustains half-way to B and half-way to C. Also, 
the point C sustains half-way to F and, on the other rafter, 
half-way to the corresponding point to F. And because these 
points of support are located at equal distances apart, there- 
fore the load on each is the same in amount. On D G make 
D a equal to 100 of any decimally divided scale, and let D a 
represent the load on 7>, and draw the parallelogram at D g. 
Then, by the same scale, D h represents {Art. 258) the pressure 
in the axis of the rafter by the load at J) ; also, JD c the pressure 
in the brace D F. Draw cd horizontal ; then D d is the ver- 
tical pressure exerted by the brace DF at F. The point F 
sustains, besides the common load represented by 100 of the 
scale, also the vertical pressure exerted by the brace DF/ 
therefore, since J) a represents the common load on D, F, or 
C, make Fe equal to the sum of Da and D d, and draw the 
parallelogram Fg ef. Then Fg^ measured by the scale, is 
the pressure in the axis of the rafter caused by the load at F^ 
and Ff is the load in the axis of the brace F B, Draw fh 
horizontal ; then Fh is the vertical pressure exerted by the 
brace F B at B. The point C, besides the common load re- 
presented by D a, sustains the vertical pressure Fh caused by 
the brace FB, and a like amount from the corresponding brace 
on the opposite side. Therefore, make Cj equal to the sum of 
D a and twice Fh, and draw j k parallel to the opposite rafter. 
Then Ok is the pressure in the axis of the rafter at C. This is 
not the only pressure in the rafter, although it is the total 
pressure at its head 0. At the point F, besides the pressure 
CJk, there is Fg. At the point D, besides these two pressures, 
there is the pressure D h. At the foot, at A, there is still an 
additional pressure : while the point D sustains the load half- 



260 . AMEBIC AIT HOrSE-CAEPENTEK. 

way to F and half-way to A^ the point A sustains the load 
half-way to D. This load is, in this case, just half the load at 
D. Therefore draw A m vertical, and equal to 50 of the scale, 
or half of D a. Extend G Atol i draw m I horizontal. Then 
4 Z is the pressure in the rafter at A caused by the weight of 
the roof from A half-way to D. I^^ow the total of the pressures 
in the rafter is equal to the sum oi Al -\- Dh f Fg added to 
Ch. Therefore make h n equal to the sum oi Al -k- Dh ^ Fg^ 
and draw no parallel with the opposite rafter, and ^ J hori- 
zontal. Then C o^ measured by the same scale, will be found 
equal to the total weight of the roof on both sides of G B. If 
D a = 100 represent the portion of the weight borne by the 
point Z), then Co^ representing the whole weight of the roof. 
should equal 600, (as it does by the scale,) for D supports just 
one-sixth of the whole load. As Cn is the total oblique thrust 
in the axis of the rafter at its foot, therefore nj is the horizon- 
tal thrust in the tie beam. 

361. — In stating the amount of pressures in the above as 
being equal to certain lines, it was so stated with the under- 
standing that the lines were simply in proportion to the weights. 
To obtain the weight represented by a line, multiply its length 
(measured by the scale used) by the load resting at D^ (or at 
i^or (7, as these are all equal in this example,) and divide the 
product by 100, and the quotient will be the weight required. 
For, as 100 of the scale is to the load it represents, so is any 
other dimension on the same scale to the load it represents. 

ZQ±— Example. Let A B {Fig. 233) equal 26 feet, GB 13 
feet, and A G 29 feet, and AG, GE, and EB, each 8f feet. 
Let the trusses be placed 10 feet apart. Then the weight oc 
Z>, for the use of the braces and rods, is, per (186), equal to 

4c7i/'6^f 8-55-f 12i|-') 
= 4 X 10 X 8i(6 X 1^ + 8-55 4- 12* x "—) 



FRAMING. 261 

= 3:1:61 X 14-398 
= 4991-3. 
This is the common load at the points D^ F^ and (7, and each 
of the lines denoting pressures multiplied by it and divided by 

4Q91*3 
100, or multiplied by the quotient of . — 49-913. the pro- 
duct will be the weight required. 49'913 may be called 50, 
for simplicity ; therefore the pressure in the brace D E equals 
112 X 50 = 5600 pounds, and in the brace F B, 140 X 50 - 
7000 pounds, and in like manner for any other strain. For 
the rafters and tie beam the total weight, as per (187), equals 

4c^(6^+8-55+6i^') 

= 4x10x52(6 xf|+8-55 + 6ix^-|) 

= 2080 X 13-148 
= 27343-68 pounds. 
This is the total weight of the roof supported by one truss. 
The oblique thrust in the rafter A G is, per (188), equal to 

ZTF_29 X 27343-68 
2 A 2x13 

== 30498-72 pounds. 

To obtain this oblique thrust geometrically : C o {Fig. 233) 
represents the weight of the roof, and measures 600 by the scale ; 
and the line G^i^ representing the oblique thrust, measures 670. 
By the proportion, 600 : 670 :: 27343-68 : 30533-8, = the 
oblique thrust. The result here found is a few pounds more 
than tlie other. This is owing to the fact that the line 6^^ is 
not exactly 670, nor is the length of the rafter precisely 29 feet. 
Were tlie exact dimensions used in each case the results would 
be identical ; but the result in either case is near enough foi 
the purpose. 

The horizontal strain is, per (189), equal to 



262 AMERICAN HOUSE- CAKPENTEE. 

Ws __ 27343- 68 X 52 
4A " 4x 13 

= 2734:3-68 pounds. 

The result gives the horizontal thrust precisely equal to the 
weight. This is as it should be in all cases where the height 
of the roof is equal to one-fourth of the span, but not other- 
wise ; for the result depends (189) upon this relation of the 
height to the span. Geometrically, the result is the same, for 
Co and nj {Fig. 233,) representing the weight and horizontal 
thrust, are precisely equal by measurement. 

363. — The weight at the head of a brace is sustained partly 
at the foot of the brace and partly at the foot of the rafter. 
The sum of tRe vertical effects at these two points is just equal 
to the weight at the head of the brace. The portion of the 
weight sustained at either point is in proportion, inversely, to 
the horizontal distance of that point from the weight ; there- 
fore, 

V=W^, (190.) 

d 

where Y equals the vertical effect at the foot of the brace ; TF, 

the weight at the head of the brace ; g, the horizontal distance 

from the foot of the rafter to the head of the brace ; and a, the 

distance from the same point to the foot of the brace. 

364. — For the oblique thrust in the brace : from the triangle 

Ff h {Fig. 233,) 

Fh : Ff:: sin. : rad. 

sin. : rad. :: V I T\ 
therefore, 

T-=Z- = y\, (191.) 

sm. h ' 

where T equals the oblique thrust in the brace ; F, the verti- 
cal pressure caused by T at the foot of the brace (190) ; a 
I and K the length and height respectively of the brace. 

365. — Example. Brace D E, Fig. 233. In this case, 
equals the product of the weight per superficial foot, m" 



FRAMING. 263 

plied by the area supported at the point Z>, equals 500C 
pounds, {Art. 362.) The length g equals 8f feet, and a equals 
ITi feet. Therefore (190), 

Y= Tr-?= 5000 X -^ = 2500 pounds 

equals ihe vertical pressure at ^ caused by the brace D E. 
Ther for the oblique thrust, I equals 9*6 feet, and h equals 4-3 
feet. Therefore, from (191), 

T=:Y\^ 2500 X ^ = 55814 pounds 
III 4'o 

equals the oblique thrust in the brace D E, In Art. 362 it 

was found to be 5600. The discrepancy is owing to like causes 

of want of accuracy in the case of the rafter, as explained in 

Art. 362. 

Another Example. — Brace FB^ Fig. 233. In this case, W 

equals the product of the weight per superficial foot, multiplied 

by the area supported by the point F^ added to the vertical 

strain caused by the brace D E. From Art. 362 the weight 

of roof on F equals 5000 pounds, and the vertical strain from 

brace D E is, as just ascertained, = 2500, total 7500, equals 

W. The length, ^, equals two-thirds of 26 feet, equals lY^-, 

and a equals 26 feet. Therefore, from (190), 

F=Tr^ = 7500 X 1^ = ^000 
a 26 

equals the vertical effect at B caused by the brace FB, 
Then, for the oblique thrust in the brace, I equals 12*2, and h 
equals 8f. Therefore, from (191) 

7 19'9 

T=Y~^ 5000 x~= Y038-5 

equals the oblique thrust or strain in the axis of the brace. It 
was 7000 by the geometrical process, {Art. 362.) 

366. — ^The strain upon the first rod, D 6r, equals simply the 
weight of the ceiling supported by it, added to the part of the 
tie beam it sustains. The weight of the tie beam will equal" 



264 



AMERICAN HOUSE-CABPENTEE. 




Fie. 283. 



FRAMING. 265 

about one pound per superficinl foot of the ceiling. The weight 
per foot for the ceiling is stated (see Art. 356 third, and 358,) 
at 9 pounds. To this add 1 pound for the tie beam, and the 
sum is 10. Then 

J^= 10 on. (192.) 

The strain on. the second tie rod equals the weight of ceiling 
supported, = iT, added to the vertical effect of the strain in the 
brace it sustains, [see (190)] or equal to 

= 10 en + V. (193.) 

The strain on the third rod is equal to iV^, added to the ver- 
tical effect of the strain in the brace it sustains, and this is the 
strain on any rod. The first rod has no brace to sustain, and 
the middle rod sustains two braces. In this case the strain 

equals 

U =^10 en + 2V. (194) 

It may be observed that V represents the vertical strain 
caused by that brace that is sustained by the rod under consi- 
deration ; and, as the vertical strain caused by any one brace is 
more than that caused by any other brace nearer the foot of 
the rafter, therefore the V of (193) is not equal to the Y of 
(194). Hence a necessity for care lest the two be confounded 
and thus cause error. 

ZQ1.—Exam2jles. The rod JD G (Fig. 233) has a strain 
which equals (192) 

JSr= 10 C72. =1 10 X 10 X 8| = 867 pounds. 

The strain on rod i^^ equals (193) 

O = 10Gn + Y=%m -{- 2500 =: 3367 pounds. 

The strain on rod G B, the middle rod, equals (194) 

TT= 10 c?i + 2 F= 867 + 2 x 5000 = 10867 pounds. 
368. — The load, and the strains caused thereby, having 
been discussed, it remains to speak of the resistance of the ma- 
terials. 

First.^ of the Rafter. — Generally this piece of timber is so 
pinioned by the roof beams or purlins as to prevent any late- 
st 



.2#6 AMEEICAN HOUSE-CAKPENTEK. 

ral movement, and the braces keep it from deflection ; there 
fore it is not liable to yield by flexure. Hence the manner of 
its yielding, when overloaded, will be by crushing at the ends^ 
or it will crush the tie beam against which it presses. The 
fibres of timber yield much more readily when pressed toge- 
ther by a force acting at right angles to the direction of their 
length, than when it acts in a line, with their length. 

The value of timber subjected to pressure in these two ways 
is shown in Art. 292, Table I., the value per square inch of 
the first stated resistance being expressed by P, and that of 
the other by G. Timber pressed in an oblique direction yields 
with a force exceeding that expressed by P^ and less than that 
by C. When the angle of inclination at which the force acts 
is just 45°, then the force will be an average between P and 
G. And for any angle of inclination, the force will vary in- 
versely as the angle ; approaching P as the angle is enlarged, 
and approaching G as the angle is diminished. It will be 
equal to G when the angle becomes zero, and equal P when 
the angle becomes 90°. The resistance of timber per square 
inch to an oblique force is therefore expressed by 

M=P^^{0-P), (195.) 

where A° equals the complement of the angle of inclination. 
In a roof, -A° is the acute angle formed by the rafter with 
a. vertical line. If no convenient instrument be at hand to 
measure the angle, describe an arc upon the plan of the 
truss — thus: with GB {Fig. 238) for radius, describe the 
arc B ^, and get the length of this arc by stepping it off with 
a pair of dividers. Then 

where a equals the length of the arc, and h equals B (7, the 
height of the roof. Therefore, 



FRAIVIING. 267 

M=F + 0-63f |((7 - P) (196.) 

equals the value of timber per square inch in a tie beam, C 
and P being obtained from Table I., Art. 292. When G for 
the kind of wood in the tie beam exceeds C set opposite the 
kind of wood in the rafter, then the latter is to be used in the 
rules instead of the former. 

369. — Having obtained the strain to which the material is 
subjected in a roof, and the capability of the material to resist 
that strain, it only remains now to state the rules for determin- 
ing the dimensions of the material. 

370. — To obtain the dimensions of the rafter : — It has been 
shown that the strain in the axis of the rafter equals (188), 

2A 
This is the strain in pounds. Timber is capable of resisting 
effectually, in every square inch of the surface pressed (196), 

P +0-631 1 ((7 -P) pounds. 

And when the strain and resistance are equal, 

P = hdlP-{-0'6S%j{C-P)l 

where h and d are respectively the breadth and depth of the 
rafter. Hence 

J d = — - — 4 • (i^'^O 

P + 0-63%~{O-P) ^ 

Example. — (Fig. 233.) The strain in the axis of the rafter 

m this example, ascertained in Art. 362, is 30498*72 pounds. 

If the timber used be white pine, then P = 300 and G= 1200, 

The length of the arc ^^ is 14J feet, and li = 13. Therefore 

M^ '-^^^^ = 32-8. 

300 4- (0-631 X '4r X 900) 

This is the area of the abutting surface at the tie beam — 

say 6 by 5^ inches. At least half this amount should be added 



268 AMERICAN HOUSE-CARPENTER. 

to allow for the shoulder, and for cutting at the joints fo* 
braces, &c. The rafter may therefore be 6 bj 9 inches. 

The above method is based upon the supposition that the 
rafter is effectually secured from flexure by the braces and 
roof beams. Sliould this not be the case, then the dimensions 
of the rafter are to be obtained by rules in Art. 298, for posts. 
IN evertheless, the abutting surface in the joint is to be deter- 
mined by the above formula (197). 

371. — To obtain the dimensions of the braces : — Usually, 
braces are so slender as to require their dimensions to be ob- 
tained by rules in Art. 298 ; the strain in the axis of the brace 
having been obtained by formula (191), or geometrically as in 
Art. 360. 

The abutting surface of the joint of the brace is to be ob- 
tained, as in the case of the rafter, by formula (195) ; A° be- 
ing the number of degrees contained in the acute angle formed 
by the brace and a vertical line, for the joint at the tie beam ; 
but for the joint at the rafter, A° is the number of degrees 
contained in the acute angle formed by the brace and a line 
perpendicular to the rafter, or it is 90, diminished by the num- 
ber of degrees contained in the acute angle formed by the 
rafter and brace. 

Examjple. — Fig. 233, Brace D E^ of white pine. In this 
brace the strain was found [Art. 362) to be 5600 pounds, the 
length of the brace is 9'6 feet. By Art. 298, the brace is 
therefore required to be 4*18 X 6 inches. For the abutting 
surface at the joints, for white pine, P equals 300 and, 6'1200. 
The angle i>J^i^ equals 63° 26'. By (197) and (195), 

T 5600 



Id 



jP + ^{0-F) 300 -f- l"-^' X (1200 - 300)] 
5600 



= 6 inches. 



934-5 

This is the area of the abuttir g surface of the joint at the tie 



Id^ 



FEAMING. 

beam. To obtain the joint at the rafter, the angle FD F. 
equals 53° 8', and hence 

T _ 6600 

TTWfi^^) ~ 300+L^F^ X (1200 - 300)] 

.= 1^ = 8-375 inches. 

300 + ('-^ X 900) 

This is the area of the abutting surface of the joint at the 
rafter. 

Another Fxainple.—BrsiGG FB, Fig. 233, of white pine, 
12-2 feet long. The strain in its axis is {Art. 362) 7000 
pounds. By Art. 298, the brace is required to be 5^ x 6 
inches. For the abutting surface of the joints, P equals 300, 
C equals 1200, and the angle I B equals 45° ; therefore, 

ld = TT = H inches. 

300 +[i x(1200-300)] 

This is the area of the abutting surface at the tie beam. For 
the surface at the rafter, the angle OFB equals 71°, and 
90 — 71 = 19, equals the angle to be used in the formula ; 
therefore, 

hd =: r^ = 14*3 inches, nearly. 

300-f [ix (1200-300)] 

This is the area of the abutting surface of the joint at the 
rafter. 

372. — To obtain the dimensions of the tie beam : — A tie 
beam must be of such dimensions as will enable it to resist 
eifectually the tensile strain caused by the horizontal thrust of 
the rafter and the cross strains arising from the weight of the 
ceiling, and from any load that may be placed upon it in the 
roof. From (17), Art. 310, 

A- ^-^ 
where S" equals the horizontal thrust, and from (189), 



270' AMERICAN HOrSE-CAEPENTEE. 

Ws 



therefore, 



^=-4X' 



B Ws 



where W equals the weight of the roof in pounds, as shown 
at (18Y) ; s, the span ; A, the height, both in feet ; and T^ a 
constant set opposite the kind of material, in Table III. ; and 
A equals the area of uncut fibres in the tie beam. About 
one-half of this should be added to allow for the requisite cut- 
ting at the joints ; or, the area of the cross section of the tie 
beam should be equal to at least f of the area of uncut fibres ; 
or, when 1) d equals the area of the tie beam, then 

T>d = %^. (198). 

Example. — ^The weight on the truss at Fig. 233 is shown to 
be {Art. 362) 2T343'68 pounds, say 27500 pounds ; the span is 
52 feet, the height 13, and the value of T for white pine is 
(Table III.) 2367, therefore 

. , 8 "^^ 8 27500 X-52 -,^, . , 
&^ = 1-^ = 1 x——-^:.174mches 

equals the area of cross section of the tie beam requisite to 
resist the tensile strain. This is smaller, as will be shown, than 
what is required to resist the cross strains, and this will be 
found to be the case generally. The weight of the ceiling is 
9 pounds per superficial foot ; the length of the longest unsup- 
ported part of the tie beam is 8f feet ; then, if the deflection 
per lineal foot be allowed at 0*015 inch, the depth of the tie 
beam will be required ((72), Table Y.) to be 6*14 inches. But 
in order efi'ectually to resist the strains tie beams are subjected 
to at the hands of the workmen, in the process of framing and 
elevating, the area of cross section in inches should be at 
least equal to the length in feet. Were it possible to guard 
against this cause of strain, the size ascertained by the rule, 6 



FEAMING. ^1 

by 6 '14:, would be sufficient ; but to resist this strain, the size 
should be 6 by 9. 

There is yet one other dimension for the tie beam required, 
and that is, the distance at which the joint for the rafter must 
be located from the end of the tie beam, in order that the 
thrust of the rafter may not split off the part against which it 
presses. This may be ascertained by Rule XI., Art. 302, for 
all cases where no iron strap or bolt is used to secure the joint ; 
but where these fastenings are used the abutment may be of 
any convenient length. And in using irons here, care should 
be exercised to have the surface of pressure against the iron 
of sufficient area to prevent indentation. 

373. — ^To obtain the dimensions of the iron suspension rods. 
By Art. 310, (17), 

A - '^ 

and T varies (Table IIL) from 5000 to 17000, according to the 
diameter inversely ; for the smaller rods are stronger in pro- 
portion than the larger ones. 

Example. — ^Taking T equal 5000, then the area of the rod 
D G {Fig. 233) requires {Art. 367) to be equal to 

corresponding to 0'469 inch diameter. This rod may be half 
inch diameter. 

Another Example.— Th^ rod FE {Fig. 233) is loaded with 
{Art. 367) 3367 pounds, therefore 

equals the area of the rod, the corresponding diameter of which 
is 0-925. This rod may be one inch diameter. 

Again, a third example; the rod C B. This rod is loaded 
with {Art. 367) 10867 pounds, therefore 



2T2 AJIERICAN HOUSE-CAEPENTEE. 

. 1086T o.^o- -u 

^ = Tooo- = '*'^'^"^^ 

equals the required area of the rod, tlie diameter correspond- 
ing tz wliich is 1*66. This rod may therefore be If inches 
diameter. 

374. — While discussing the principles of strains in roofs and 
deducing rules therefrom, the truss indicated in M.g. 233 has 
been examined throughout. The result is as follows : rafter, 
6x9; tie beam, (6 X 6, or) 6x9; the first brace from the 
wall, 4 J X 6 inches, with an abutting surface at the lower end 
of 6 inches, and at the upper end of 8f inches ; the other 
brace, 5J x 6 inches, with an abutting surface at the lower 
end of 9^ inches, and at the upper end of 14y^o inches; the 
shortest rod, -J inch diameter ; the next, 1 inch diiameter ; and 
the middle rod, If inches diameter. 

PRACTICAL RULES AND EXAMPLES. 

JF'or Roofs Loaded as per Art. 356. 

375. — Rule LIII. To obtain the dimensions of the rafter 
Multiply the value of R (Table IX., Art 376) by the span 
of the roof, by the length of the rafter, and by tlie distance 
apart from centres at which the roof trusses are placed, all in 
feet, and divide the product by the sum of twice the height 
of the roof multiplied by the value of P, Table I., set opposite 
the kind of wood used in the tie beam, added to the difference 
of the values of C and P in said table multiplied by 1:^ times 
the length of the arc that measures the acute angle formed 
between the rafter and a vertical line, the arc having the height 
of the roof for radius (see arc B G^ Fig. 233), and the quo 
tient will be the area of the abutting surface of the joint at 
the foot of the rafter. To the abutting surface add its half, 
and the sum will be the area of the cross section of the rafter 



FRAMING. 273 

This rule is upon the presumption that the rafter is secured 
from flexure by the roof beams and by braces and ties at 
short intervals, as in Fig. 233. In roofs where the rafter does 
not extend up to the ridge of the roof but abuts against a 
horizontal straining beam (c, Fig. 237), in the rule for rafters, 
take for the length of the rafter the distance from the foot of 
the rafter to the ridge of the roof; or, a distance equal to 
what the rafter wcmld be in the absence of a straining beam. 
The area of cross section of the straining beam should be made 
equal to that of the rafter, as found by the rule so modified. 

Example. — Find the dimensions of a rafter for a roof truss 
whose span is 52 feet, and height 13 ; the length of the rafter 
being 29 feet, the trusses placed 10 feet apart from centres, 
and the arc measuring the angle at the head of the rafter 
(having the height of the roof for radius) being 14;^ feet, 
white pine being used in the tie beam. The height of this 
roof being in proportion to the span as 1 to 4, the value of H 
in Table IX. is 52*6 ; multiplying this, in accordance with the 
rule, by 52, the .span of the roof, and bj 29, the length of the 
rafter, and by 10, the distance between the roof trusses, the 
product is 793208. The value of F for white pine in Table I. 
is 300 ; multiplying this by 2 x 13 = 26, twice the height of 
the roof, the product is 7800. The value of C for white pine, 
(Table I.) is 1200, hence the difference of the values of G and 
F is 1200-300 = 900; this multiplied by U, and by 14i, 
the length of the arc, the product is 16031 ; this added to the 
7800 aforesaid, the sum is 23831. The aforesaid product of 
793208, divided by this 23831, the quotient, 33*3, equals the 
area in inches of the abutting surface of the joint at the tie 
beam. To this add 16*7, its half, and the sum, 50, equals the 
area of cross section of the rafter. This divided by the thick- 
ness of the rafter, say 6 inches, the quotient, 8§, is the breadth. 
The rafter is therefore to be 6 X 8^ inches. It may be made 
6x9, avoiding the fractions. 

35 



274 



AMERICAN HOJSE-CARrENTEK, 



BTQ. — ^The following table, calculated npon data in Art. 358. 
presents the weight per foot for roofs of various inclinations, 
and covered with slate. 



TABLE IX. 





The vertical strain per foot of surface supported, measured horizontally. 


When height of roof 






is to span as 










on rafters = ^K = 


on braces 


= Q- 


1 to 8 


48 pounds 


49-5 pc 


unds. 


1 " 7 


48-6 " 


60-6 


<( 


1 " 6 


49-4 " 


61-9 


« 


1 " 6 


50-6 " 


54- 


« 


1 " 4 


62-6 " 


57-6 


<« 


1 '* 3 


56-3 " 


64- 


i( 


1 " 2 


62-1 *' 


76-2 


« 


1 « 1 


81- " 


101- 


« 



To get the proportion that the height bears to the span, di- 
vide the span by the height ; then unity will be to the quotient 
as the height is to the span. In case the quotient is not a 
whole number, the required value of i? or § will not be found 
in the above table, but maybe obtained thu&: multiply the 
decimal part of the quotient by the difference of the values of 
B set opposite the two proportions, between which the given 
proportion occurs as an intermediate, and subtract the product 
from the larger of the two said values of i? ; the remainder 
will be the value of B required. The process is the same for 
the values of Q. 

Example. — A roof whose span is 60 feet, has a height of 25 
feet. Then 60 divided by 25 equals 2 -4. The proportion, 
therefore, between the height and span is 1 to 2'4. This pro- 
portion is an intermediate between 1 to 2 and 1 to 3. The 
values of R^ opposite these two, are 63*7 and 56*3. The dif- 
ference between these values is 7*4 ; this multiplied by 0*4, the 
decimal portion of the quotient, equals 2*96 ; this subtracted 
from 63*7, the larger value of ^, the remainder, 60-74, is the 
required «othie of R. 



FRAMING. 275 

The values of R and Q are those for a roof covered with 
slate weighing Y pounds per superficial foot of the roof sur- 
face. When the roof covering is either lighter or heavier, sub- 
tract from or add to the table values, the difference of weight 
between 7 pounds and the weight of the covering used, and 
the remainder, or sum, will be the value of i? or Q required. 

377. — Rule LIY. To obtain the dimensions of braces. 

« 

Multiply the value of Q (Table IX., Art. 376) by the distance 
apart in feet at which the roof trusses are placed, and by the 
horizontal distance in feet from a point half-way to the next 
point of support of the rafter on one side of the brace, to a 
corresponding point on the other side. The product -will be 
the weight in pounds sustained at the head of the brace. To 
this add the vertical strain {Art, 360) on the suspension rod 
located at the head of the brace, and make a vertical line 
dropped from the head of the brace, as F e^ Fig. 233, equal, 
by any convenient scale, to this sum, and draw the parallelo- 
gram Ff eg. Then Ff.^ measured by the same scale, equals 
the pressure in the axis of the brace F B. Multiply this pres- 
sure in pounds by the square of the length of the brace in feet, 
and divide the product by the breadth of the brace in inches 
multiplied by the value of ^ (Table II., Art 293). The cube 
root of the quotient will be the thickness of the brace in 
inches. If this cube root should exceed the hreadth of the 
brace, the result is not correct, and the calculation will have to 
be made anew, taking a larger dimension for the breadth. 

Examjple. — The brace F B {Fig. 233) is of white pine, and 
is required to sustain a pressure in its axis of 7000 pounds 
{Art. 362). The length of the brace is 12 feet and its breadth 
6 inches, what must be its thickness ? Here 7000, the pressure, 
multiplied by 144, the square of the lengtJi, equals 1008000. 
The value of B is 1175 ; this by 6, tlie breadth of the brace, 
equals 7050. The product 1008000 divided by the product 
7050 equals 143. the cube root of which, 5*23, is the required 



276 A.MEIIICAN HOUSE- CARPENTER. 

thickness of the brace in inches. Tlie brace will therefore be 
6-23 bj 6 inches, or 5^ by 6. 

378. — Rule LY. To obtain the area of the abutting sur 
face of the ends of braces. Divide the number of degrees 
contained in tlie complement of the angle of inclination by 90, 
and multiply the quotient by the difference of the values of C 
and _P, set opposite the kind of wood in the tie beam or rafter, 
in Table L, Art. 292 ; and to the product add the said value 
of -P, and by the sum divide the pressure in the axis of the 
brace, and the quotient will be the area of the abutting sur- 
face. 

The complement of the angle of inclination referred to is, 
for the foot of the brace, the acute angle contained between 
the brace and a vertical line ; and for the head of the brace, 
the acute angle contained between the brace and a line per- 
pendicular to the 'rafter. 

Example. — To find the abutting surface of the ends of the 
brace F B {Fig. 233). The complement of the angle of incli- 
nation, for ih.Q foot of the brace, is that contained between the 
lines F B and F E^ and measures by the protractor, 45°. The 
tie beam is of white pine, and the values of P and C for this 
wood are 300 and 1200 respectively, and the pressure in tlie 
axis of the brace is 7000 pounds. Now by the rule, 45 di- 
vided by 90 equals 0*5, this by the 900, the difference of the 
values of C and P^ equals 450 ; to this add 300, the value of 
jP, and the sum is 750. The pressure in the axis of the brace, 
7000, divided by this 750, equals 9§, the required area of the 
abutting surface at the foot of the rafter. The complement of 
the angle of inclination for the head of the brace is that con- 
tained between the lines B F and i^j?, and measures by the 
protractor 19°. The rafter being of white pine, the values of 
jP and C are as before. By the rule, 19 divided by 90 equals 
0'2^, and this multiplied by 900, the difference of the values 
of P and (7, equals 190 ; to this add 300, the value ^'iP, and 



FKAMING. 277 

the sum is 490. The pressure, 7000, divided by this 490, 
eqnals 14'3 inches, the required area of the abutting surface at 
the head of the brace. 

379. — To obtain the dimensions of the tie beam. Tie beams 
are subjected to two kinds of strain — tensile and transverse. 

Rule LYI. — To guard against the tensile strain, multiply the 
value of R (Table IX., Art. 376) by three times the distance 
apart at which the trusses are placed, and by the squai-e of the 
span of the truss, both in feet. Divide this product by the 
value of T, (Table III., Art. 308) set opposite the kind of wood 
in the tie beam, multiplied by 8 times the height of the roof 
in feet, and by the breadth of the tie beam in inches. The 
quotient will be the required depth in inches. 

The result thus obtained is usually smaller than that re- 
quired to resist the cross strain to which the tie beam is sub- 
jected. The dimensions required to resist this strain, where 
there is simply the weight of the ceiling to support, may be 
obtained by this rule : 

Rule LYII. — Multiply the cube of the longest unsupported 
part of the tie beam by 400 times the distance apart at which 
the trusses are placed^ both in feet ; and divide the product by 
the breadth of the tie beam in inches, multiplied by the value 
of E^ (Table IL, Art. 293) set opposite the kind of wood in the 
tie beam, and the cube root of the quotient will be the re- 
quired depth of the tie beam in inches. 

The result thus obtained may not be sufficient, in some cases, 
to resist the strains to which the tie beam is subjected in the 
hands of the workmen during the process of framing. 

Rule LYIII. — To resist these strains the area of cross sec- 
tion in inches should be at least equal to the length in feet. 

Example. — The tie beam in Fig. 233. For this case we have 
the value of R 52*6, the trusses placed 10 feet from centres, 
the span 52 feet, the height 13 feet, the breadth 6 inches, and 
<Jie value of T 2367. Then by the rule, 52-6 X 3 x 10 x 52' 



278 AlklERICAN HOUSE-CAEPENTER. 

=.- 4206912, and 2367 x 8 x 13 x 6 - 1477008 ; the formei 
product divided by the latter, the quotient equals 2-9, equals 
the required depth of the tie beam in inches. The other 
strains will require the depth to be more. To resist the cross 
strains, we have the longest unsupported part of the tie beam 
8f feet, (this dimension is frequently greater than this,) distance 
from centres 10 feet, and breadth 6 inches. Then, by the rule, 
8f 3 X 400 X 10 = 2603852, and 6 x 1750 = 10500 ; the former 
product divided by the latter, the quotient is 248, the cube root 
of which, 6*28, equals the required depth in inches. The tie 
beam therefore is to be 6 by 6'28 inches, or 6 x 7 inches. But 
if not guarded against severe accidental strains from careless 
handling this size would be too small. It would, in this case, 
require to be 52 inches area of cross section, say 6x9 inches. 

380. — To obtain the diameter of the suspension rods, when 
made of o-ood wrouo;ht iron. 

Rule LIX. — Divide the weight or vertical strain, in pounds, 
by 4000. The square root of the quotient will be the required 
diameter of the rod in inches. 

Example. — A suspension rod is required to sustain 16000 
pounds, what must be its diameter ? Dividing by 4000, the 
quotient is 4 ; the square root of which, 2, is the required 
diameter. 

The vertical strain on any rod is equal to the weight of so 
much of the ceiling as is supported by the rod, added to the 
vertical strain caused by each brace that is footed in the tie 
beam at the rod. The weight of the ceiling supported by a 
rod, is equal to ten times the distance apart in feet at which 
the trusses are placed, multiplied by half the distance in feet 
between the two next points of support, one on either side of 
the rod. The vertical strain caused by the braces can be as 
certained geometrically, as in Art. 360. 

381. — When the suspension rods are located as in Fig. 233, 
dividing the span into equal } arts, the diameter ol the rods 



FRAMING. 279 

may be obtained without the preliminary calculation of the 
strain, as follows : 

Riile LX. — For the first rod from the wall. Multiply the 
distance apart at which the trusses are placed by the distance 
apart between the suspension rods, and divide the product by 
400. The square root of the quotient will be the required 
diameter of the rod. 

Example. — Eod 1) G^ Fig. 233. In this figure the rods are 
located at 8f feet apart, and the distance between the trusses 
is 10 feet. Therefore, 10 x 8| = 86? ; this divided by 400, 
the quotient is 0*2167, the square root of which, 0*465, is the 
required diameter. The diameter may be half an inch. 

Bule LXI. — For the second rod from the wall. To the 
value of Q (Table IX., Art. 376) add 20, and multiply the sum 
by the distance apart at which the trusses are placed and by 
the distance between the rods, both in feet, and divide the pro- 
duct by 8000. The square root of the quotient will be the re- 
quired diameter. 

Example. — Rod F E^ Fig. 233. The distances apart in this 
case are as stated in last example. The value of Q is 57*6, and 
when added to 20 equals 77-6. Therefore, 77*6 X 10 x 8| == 
6673| ; this divided by 8000, the quotient is 0*8341, the square 
root of which, 0*91, is the required diameter. This rod may 
be one inch diameter. 

Bide LXIL — For the centre rod. To the value of Q (Table 
IX., Art. 376) add 5, and multiply the sum by the distance 
apart at which the trusses are placed and by the distance apart 
between the rods, both in feet, and divide the product by 2000. 
The square root of the quotient will equal the required diameter. 

Example. — Rod G B, Fig, 233. The distances apart as be- 
fore, and the value of Q the same. To Q add 5, and the sura 
is 62*6. Then 62*6 x 10 x 81 == 5425J ; this divided by 2000, 
the quotient is 2*7126, the square root of which, 1*647, equal? 
the required diameter. This rod may be If inches diameter. 



280 AMERICAN nOUSE-CAEPENTEK. 

382. — For all wrought iron straps and bolts the dimensions 
may be found by this rule. 

Rule LXIII. — Divide the tensile strain on the piece, in 
pounds, by 5000, and the quotient will be the area of cross 
section of the required bar or bolt, in inches. 

383. — Roof-beams, jack-rafters, and purlins. All pieces of 
timber subject to cross strains will sustain safely much greater 
strains when extended in one piece over two, three, or more 
distances between bearings; therefore roof-beams, jack-rafters, 
and purlins should, if possible, be made in as long lengths as 
practicable ; the roof-beams and purlins laid on, not framed 
into, the principal rafters, and extended over at least two 
spaces, the joints alternating on the trusses; and likewise the 
jack-rafters laid on the purlins in long lengths. The dimen- 
sions of these several pieces may be obtained by the following 
rule : 

Rule LXIY.— From the value of Q (Table IX., Art. 376) 
deduct 10, and multiply the remainder by 33 times the distance 
from centres in feet at which the pieces are placed, and by the 
cube of the distance between bearings in feet ; divide the pro- 
duct by the value of E (Table IL, Art. 293) for the kind of 
wood used and extract the square root of the quotient. The 
square root of this square root will be the required depth in 
inches. Multiply the depth thus obtained by the decimal 0-6, 
and the product will be the required breadth in inches. 

Examjple. — Roof-beams of white pine placed 2 feet from cen- 
tres, resting on trusses placed 10 feet from centres, the height 
and the span of the roof being in proportion as 1 to 4. In 
this case the value of Q is 57*6. By the rule, 57*6 — 10 = 47*6, 
and 47-6 x 33 X 2 x lO'' = 3141600. This iivided by 1750, 
the quotient is 1795-2, the square root of which is 42-37, and the 
square root of 42-37 is 6-5, the required depth. This multi- 
plied by 0-6 equals 3*9, the required breadth. These roof 
beams may therefore be 4 by 6J inches. 



FEAMING. 



281 



384-. — Five examples of roofs are shown at Figs. 234, 235, 
236, 237, and 238. In Fig. 234, a is an iron suspension rod, 
h h are braces. In Fig. 235, 
«, <z, and 5 are iron rods, and 
dd^c c, are braces. In Fig. 
236, ah are iron rods, (^<^ jT 
braces, and c the straining — ' 
beam. In i^^'^. 237, aa^hh^ 






are iron rods, ee^ dd^ are braces, and c is a straining beam. 
In Fig. 238, purlins are located at P P^ &c. ; the inclined beam 
that lies upon them is the jack-rafter; the post at the ridge is 
the king post, the others are queen posts. In this design the tie 
beam is increased in height along the middle by a strengthen- 
ing piece [Ah. 348), for the purpose of sustaining additional 
weight placed in the room formed in the truss. 

385. — Fig. 239 shows a method of constructing a truss having 
a huilt-rih in the place of principal rafters. The proper form 

for the curve is that of a parabola, {Aj^. 127.) Tliis curve, 

86 



AMERICAN HOUSE-CAEPENTEE. 



when as flat as is described in the figure, approximates so 
near to that of the circle, that the hitter may be used in its 
stead. The height, a h, is just half of a c^ the curve to pass 

through the middle of the rib. 
The rib is composed of two series 
of abutting pieces, bolted toge- 
ther. These pieces should be as 
long as the dimensions of the tim- 
ber will admit, in order that there 
may be but few joints. The sus- 
pending pieces are in halves, 
notched and bolted to the tie- 
beam and rib, and a purlin is 
framed upon the upper end of 
each. A truss of this construc- 
tion needs, for ordinary roofs, no 
^ .^ diagonal braces between the sus- 
pending pieces, but if extra 
strength is required the braces 
may be added. The best place 
for the suspending pieces is at the 
joints of the rib. A rib of this 
kind will be sufficiently strong, 
if the area of its section contain 
about one-fourth more timber, 
than is required for that of a raf- 
ter for a roof of the same size. 
The proportion of the depth to 
the thickness should be about as 
10 is to 7. 
386 — Some writers have given designs for roofs similar to 
Fig. 240, having the tie-beam omitted for the accommodation 
of an arch in the ceiling. This and all similar designs are se- 




FRAMING. 



283 



riously objectionable, and should always be avoided ; as the 
small height gained bj the omission of the tie-beam can never 




Fig. 240. 



compensate for the powerful lateral strains, which are exerted 
by the oblique position of the supports, tending to separate the 



284: 



AArFKICAX HOrSE-CAEPEXTEE. 



walls. Where an arch is required in the ceiling, the hest plan 
is to carry up the walls as high as the top of the arch. Then, 
by using a horizontal tie-beam, the obliqne strains will be en- 
tirely removed. Many a public building, by my own obser- 
vation, has been all but ruined bv the settlins: of the roof, 
consequent upon a defective plan in the formation of the truss 
in this respect. It is very necessary, therefore, that the hori- 
zontal tie-beam be used, except where the walls are made so 
strong and firm by buttresses, or other support, as to prevent 
a possibility of their separating. 




Fig. 241. 



387. — Fig. 24:1 is a method of obtaining the proper lengths and 
bevils for rafters in a hip-roof: a h and h c are walls at the angle 
Df the building ; 5 e is the seat of the hip-rafter and gf of a 
jack or cripple rafter. Draw e A, at right angles to 5 ^, and make 
it equal to the rise of the roof; join h and A, and A h will be the 
length of the hip-rafter. Tlirough e^ draw d i, at right angles to 
he ; upon 5, with the radius, hh, describe tlie arc, A «', cutting dt 
in i ; join h and i, and extend gf to meet h i mj ; then gj will 



FRAMING. 



285 



be the length of the jack-rafter. The length of each jack- rafter is 
found in the same manner — by extending its seat to cut the line, 
h i. From/, draw f k, at right angles iofg^ also//, at right 
angles to be; make/ ^^ equal to // by the arc, / k, or make g k 
equal to g-j by the arc,^* k ; then the angle at^" will be the top 
hevil of the jack-rafters, and the one at k will be the down-bevil.^ 
388. — To find the hacking of the hip-rafter. At any con 
venient place in h e, [Fig. 241,) as o, draw m n, at right angles to 
be; from o, tangical to b A, describe a semi-circle, cutting 6 e in 
s ; join rti and s and n and s ; then these lines will form at s the 
proper angle for beviling the top of the hip-rafter. 

DOMES.t 




Fig. 242. 




Fig. 248. 

* The lengths and bevils of rafters for xooi-valleys can also be found by ne above 
process t See also Art. 237. 



286 



AMERICAN HOUSE-CARPENTER. 



389. — The most usual form for domes is that of the sphere, the 
base bemg circular. Wlien the interior dome does i ot rise toe 
high, a horizontal tie may be thrown across, by which any de- 
gree of strength required may be obtained. Fig. 242 shows a 
section, and Fig. 243 the plan, of a dome of this kind, a h being 
the tie-beam in both. Two trusses of this kind, {Fig. 242,) pa- 
rallel to each other, are to be placed one on each side of the open- 
ing in the top of the dome. Upon these the whole framework is to 
depend for support, and their strength must be calculated accord- 
ingly. (See the first part of this section, and Art. 356.) If the 
dome is large and of importance, two other trusses may be intro- 
duced at right angles to the foregoing, the tie-beams being pre- 
served in one continuous length by framing them high enough to 
pass over the others. 




Fig. 244. 




Fii. 245. 



390. — When the interior dome rises too high to admit of a leve. 



FRAMING. 



287 



tie-beam, the framing may be composed of a succession of ribs 
standing upon a continuous circular curb of timber, as seen at 
Fig. 244 and 245,— the latter being a plan and the former a sec- 
tion. This curb must be well secured, as it serves in the place 
of a tie-beam to resist the lateral thrust of the ribs. In small 
domes, th^se ribs may be easily cut from wide plank ; but, where 
an extensive structure is required, they must be built in two 
thicknesses so as to break joints^ in the same manner as is descri- 
bed for a roof at Art. 885. They should be placed at about two 
feet apart at the base, and strutted as at a in Fig. 244. 

391. — The scantling of each thickness of the rib may be as 
follows : 

For domes of 24 feet diameter, 1x8 inches. 
" " 36 " lixlO " 

« ' 60 " 2x13 " 

" " 90 " 2|xl3 " 

« " 108 " 3x13 " 

392. — Although the outer and the inner surfaces of a dome 
may be finished to any curve that may be desired, yet the framing 
should be constructed of such a form, as to insure that the cwve 
of equilibrium will pass through the middle of the depth of the 
framing. The nature of this curve is such that, if an arch or 
dome be constructed in accordance with it, no one part of the 
structure will be less capable than another of resisting the strains 
and pressures to which the whole fabric may be exposed. The 
curve of equilibrium for an arched vault or a roof, where the load 
is equally diffused over the whole surface, is that of a parabola, 
{Art. 127 ;) for a dome, having no lantern^ tower or cupola above 
it, a cubic parabola, {Fig. 246 ;) and for one having a tower, &c., 
above it, a curve approaching that of an hyperbola must be adopted, 
as the greatest strength is required at its upper parts. If the 
curve of a dome be circular, (as in the vertical section. Fig. 244,) 
ihe pressure will have a tendency to burst the dome outwards at 
about one-third oi its height. Therefore, when this form is used 



288 



AMERICAN HOUSE CARPENTER. 



in the construction of an extensive dome, an iron band should be 
placed around the framework at that height ; and Avhatever ma} 
be the form of the curve, a bond or tie of some kind is necessary) 
around or across the base. 

If the framing be of a form less convex than the curve of 
equilibrium, the weight will have a tendency to crush the ribs in- 
wards, but this pressure may be effectually overcome b^^ strutting 
between the ribs ; and hence it is important that the struts be so 
placed as to form c ntinuous horizontal circles. 








^^^-'"^^ 







/^ 










/ 






^ - 











/ 










f>i 


/ 




1 






o/ 






1 






X 















a -) I 



S J 



e I 

Fig. 246. 



393. — To describe a cubic parabola. Let a b] {Fig. 246,) be 
the base and b c the height. Bisect « 6 at c?, and divide a d into 
100 equal parts; of these give d e 26, ef 18^,/ g 14^, g h 12^, 
h i lOf, ij 9i, and the balance, 8|, to^" a; divide b c into 8 equal 
parts, and, from the points of division, draw lines parallel to a 6, 
to meet perpendiculars from the several points of division in a ft, 
at the points, o, o, o, &c. Then a curve traced through these 
points will be the one required. 

394. — Small domes to light stairways, &c., are frequently made 
elliptical in both plan and section ; and as no two of the ribs in 
one quarter of the dome are alike in form, a method for obtaining 
the curves is necessary. 

395. — To find the curves for the ribs of an elliptical dome 
Let abed, [Fig. 247,) be the plan of a dome, and e f the seat 



FRAMING. 



289 




247. 



or one of the ribs. Then take e f for the transverse axis ana 
twice the rise, o g^ of the dome for the conjugate, and describe 
(according to Art, 115, 116, (fee.,) the semi-ellipse, e ^/, which 
will be the curve required for the rib, e g f. The other ribs are 
found in the same manner. 



h 4 




Fig. 248. 



396. — To find the shape of the covering for a spherical 
dome. Let .4, [Fig. 248,) be the plan and B the section of a 
given dome. From a, draw a c, at right angles to a 6 ; find the 
stretch-out, {Art. 92,) of o 6, and make d c equal to it ; divide the 
arc, 6, and the line, d c, each into a like number of equal parts, 

37 



290 



AMERICAN HOUSE-CARPENTER. 



as 5, (a large number will insure greater accuracy than a small 
one ;) uponc, through the several points of division in c d^ describe 
the arcs, o o? o, 1 e 1, 2/ 2, (fee. ; make d o equal to half the width 
of one of the boards, and draw o s, parallel to a c ; join s and a^. 
and from the points of division in the arc, o b, drop perpendicu- 
lars, meeting a 5 in ij k I ; from these points, draw i 4, j 3, (fee, 
parallel to a c; make d o, e I, (fee, on the lower side of a c, equal 
to d 0, e 1, (fee, on the upper side ; trace a curve through the 
points, 0, 1, 2, 3, 4, c, on each side of c? c ; then o c o will be 
the proper shape for the board. By dividing the circumference of 
the base, A, into equal parts, and making the bottom, o d o,of the 
board of a size equal to one of those parts, every board may be 
made of the same size. In the same manner as the above, the 
•hape of the covering for sections of another form may be found, 
such as an ogee, cove, (fee. 




397. — To find the curve of the boards when laid in horizon- 
tal com^ses. Let ABC, {Fig. 249,) be the section of a given 
dome, and D B its axis. Divide B C into as many parts as 
there are to be courses of boards, in the points, 1, 2, 3, (fee. ; 
through 1 and 2, draw a line to meet the axis extended at a . 
then a will be the centre for describing the edges of *he board, B. 
Through 3 and 2, draw 36; then b will be the centre for describing 
F. Through 4 and 3, draw Ad; then d will be the centre for G, 
B is the centre for the arc, \ o. If this method is taken to i^ii^ 



FRAMING. 



291 



the centres for the boards at the base of the dome, they would 
occur so distant as to make it impracticable : the following metl od 
is preferable for this purpose. G being the last board obtained by 
the above method, extend the curve of its inner edge until it 
merts the axis, D B, in e ; from 3, through e, draw 3/, meeting 
the arc, A jB, in/; join /and 4, /and 5 and/ and 6, cutting the 
axis, D B, in s, n and m ; from 4, 5 and 6, draw lines parallel to 
A Cand cutting the axis in c,p and r; make c 4, {Fig, 250,) 




equal to c 4 in the previous figure, and c s equal to c 5 also in the 
previous figure ; then describe the inner edge of the board, H^ 
according to Art. 87 : the outer edge can be obtained by gauging 
from the inner edge. In like manner proceed to obtain the next 
board — taking p 5 for half the chord and p n for the height of the 
segment. Should the segment be too large to be described 
easily, reducf; it by finding intermediate points in the curve, as at 
Art 86. 




398. — To find the shape of the angle-rib for a polygonal 
di^me. Let A G H, {Fig. 251,) be the plan of a given dome, and 



292 



AMERICAN HOUSE-CARPENTER. 



C Z) a vertical section taken at the line, ef. From 1, 2, 3, &c. 
in the aic, C D, draw ordinates, parallel to A D, to meet/ G , 
from the points of intersection on / G, draw ordinates at right- 
angles to/ G ; make s 1 equal to o 1, 5 2 equal to o 2, &c. ; then 
GfB^ obtained in this way, will he the angle-rib required. The 
best position for the sheathing-boards for a dome of this kind is 
horizontal, but if they are required to be bent from the base to 
the vertex, their shape may be found in a similar manner to that 
shown at Fig. 248. 

BRIDGES. 

399. — Various plans have been adopted for the construction of 
bridges, of which perhaps the following are the most useful. 
Fig. 252 shows a method of constructing wooden bridges, where 
the banks of the river are high enough to permit the use of the 
tie-beam, a h. The upright pieces, c c?, are notched and bolted 
on in pairs, for the support of the tie-beam. A bridge of this 
construction exerts no lateral pressure upon the abutments. This 
method may be employed even where the banks of the river are 
low, by letting the timbers for the roadway rest immediately upon 
the tie-beam. In this case, the framework above will serve the 
purpose of a railing. 




Fig. 252. 



400. — Fig. 253 exhibits a wooden bridge without a tie-beam. 
Where staunch buttresses can be obtained, this method may be 
recommended ; but if there is any doubt of their stability, it 



FRAMING. 



293 




Fig. 253. 



should not be attempted, as it is evident that such a system of 
framing is capable of a tremendous lateral thrust. 




Fig 254 



401. — Fiior. 254 represents awooden bridge in which a built-? ib^ 
(see Art. 385,) is introduced as a chief support. The curve ot 
equilibrium will not differ much from that of a parabola : this, 
therefore, may be used — especially if the rib is made gradually a 
little stronger as it approaches the buttresses. As it is desirable 
that a bridge be kept low, the following table is given to show the 
least rise that may be given to the rib. 



Span in feet. 


Least rise in feet. 


Span in feet. 


Least rise iu fett. 


Span in feet. 


Least rise in feet. 


30 


0-5 


i 120 


7 


280 


24 


40 


0-8 


140 


8 


300 


23 


50 


1-4 


160 


10 


320 


32 


60 


2 


180 


11 


.350 


39 


70 


n 


200 


12 


380 


47 


1 80 


3 


220 


14 


400 


53 


90 


4 


240 


17 






100 


5 


260 


20 







The rise should never be made less than this, but in all cases 



294 



AMERICAN HOUSE-CARPENTIiiR. 



greater if practicable ; as a small rise requires a greater quantity 
of timber to make the bridge equally strong. The greatest uni- 
form weight with which a bridge is likely to be loaded is, proba- 
bly, that of a dense crowd of people. This may be estimated at 
66 pounds per square foot, and the framing and gravelled road- 
way at 234 pounds more ; which amounts to 300 pounds on a 
square foot. The following rule, based upon this estimate, may 
be useful in determining the area of the ribs. Iiule LXY. — 
Multiply the w^idth of the bridge by the square of half the span, 
both in feet ; and divide this product by the rise in feet, multi- 
plied by the number of ribs ; the quotient, multiplied by the 
decimal, O'OOll, will give the area of each rib in feet. When 
the roadway is only planked, use the decimal, O'OOOT, instead of 
O'OOll. Example. — What should be the area or the ribs for a 
bridge of 200 feet span, to rise 15 feet, and be 30 feet wide, w^ith 
3 curved ribs ? The half of the span is 100 and its square is 
10,000; this, multiplied by 30, gives 300,000, and 15, muUi- 
phed by 3, gives 45 ; then 300,000, divided by 45, gives 6666f, 
which, multiplied by O'OOll, gives 7'333 feet, or 1056 inches for 
the area of each rib. Such a rib may be 24 inches thick by 44 
inches deep, and composed of 6 pieces, 2 in width and 3 in depth. 




Fig. 255. 



. 402. — The above rule gives the area of a rib, that would be re- 
quisite to support the greatest possible uniform load. But ir. 
large bridges, a variable load, such as a heavy wagon, is capable 
of exerting much greater strains ; in such cases, therefor*?, the 
rib should be made larger. The greatest concentrated load a 



FRAMING, 295 

bridge will be likely to encounter, may be estimated at from about 
20 to 50 thousand pounds, according to the size of the bridge. 
This is capable of exerting the greatest strain, when placed at 
about one-third of the span from one of the abutments, as at h 
{Fig. 255.) The weakest point of the segment, b g" c, is at g, 
the most distant point from the chord line. The pressure exerted 
at b by the above weight, may be considered to be in the direction 
cf the chord lines, b a and be; then, by constructing the paral- 
lelogiaT. of forces, e b f d^ according to Art. 258, b f will show 
the pressure in the direction, b c. Then the scantling for the rib 
may be found by the following rule. 

Jiule LXVl. — Multiply the pressure in pounds in the direc- 
tion h c, by the distance g A, and by the square of the distance 
h G, both in feet ; and divide the product by the united breadth 
in inches of the several ribs, multiplied by the value of £ 
(Table 11. , Art. 293) for the kind of wood used ; and the cube 
root of the quotient will be the required depth of the rib in 
inches. 

Example. — A bridge is to have three white pine ribs each 
20 inches wide ; the pressure in the direction b c, {Fig. 255) 
is equal to 60,000 pounds, the distance b c equals 60 feet, and 
the distance g h equals 10 feet. What must be the depth of 
the ribs, the value of B (Table II.) being for white pine 1175 ? 
Here, by the rule, 60,000 X 10 x 60'^ = 2,160,000,000. Then 
1175 X 3 X 20 =3 70,500. The former product divided by the 
latter equals 30,638, the cube root of which, 31-29, equals 
the required depth in inches. The ribs are, therefore, to be 20 
by 31 J- inches. 

403. — In constructing these ribs, if the span be not over 50 
feet, each rib may be made in two or three thicknesses of timber, 
(three thicknesses is preferable,) of convenient lengths bolted 
together ; but, in larger spans, where the rib will be such as to 
render it difficult to procure timber of sufficient breadth, they 
may bo coiistracted by bending the pieces to the proper curve 



296 



AMERICAN HOUSE-CARPENTER. 



md bolting them together. In this case, where timber of suffi 
cient length to span the opening cannot be obtained and scarfing 
is necessary, such joints must be made as will resist both tension 
and compression, (see Fig. 264 ) To ascertain the greatest depth 
for the pieces which compose the rib, so that the process of bend 
ing may not injure their elasticity, multiply the radius of curvature 
in feet by the decimal, 0*05, and the product will oe the depth in 
inches. Example. — Suppose the curve of the rib to be described 
with a radius of 100 feet, then what should be the depth ? The 
radius in feet, 100, multiplied by 0-05, gives a product of 5 inches. 
White pine or oak timber, 5 inches thick, would freely bend tc 
the above curve ; and, if the required depth of such a rib be 2( 
inches, it would have to be compos-ed of at least 4 pieces. Pitch 
pine is not quite so elastic as white pine or oak — its thickness 
may be found by using the decimal, 0*046, instead of 0*05. 




Fig. 256. 



404. — When the span is over 250 feet, sl framed rib, formed a,s 
in Fig, 256, would be preferable to the foregoing. Of this, the 
upper and the lower edges are formed as just described, by bend- 
ing the timber to the proper curve. The pieces that tend to the 
centre of the curve, called radials, are notched and bolted on in 
pairs, and the cross-braces are halved together in the middle, and 
abut end to end between the radials. The distance between the 
ribs of a bridge should not exceed about 8 feet. The roadway 



FRAMING. 297 

should be suj ported by vertical standards bolted to the ribs ai 
about every 10 to 15 feet. At the place where they rest on the 
ribs, a double, horizontal tie should be notched and bolted on the 
back of the ribs, and also another on the under side ; and diago- 
na ' braces should be framed between the standards, over the space 
between the ribs, to prevent lateral motion. The timbers for the 
roadway may be as light as their situation will admit, as all use- 
less timber is only an unnecessary load upon the arch. 

405. — It is found that if a roadway be 18 feet wide, two car- 
riages can pass one another without inconvenience. Its width 
therefore, should be either 9, 18, 27 or 36 feet, according to the 
amount of travel. The width of the foot-path should be 2 feet 
for every person. When a stream of Avater has a rapid current, 
as few piers as practicable should be allowed to obstruct its 
course ; otherwise the bridge will be liable to be swept away by 
freshets. When the span is not over 300 feet, and the banks of 
the river are of sufficient height to admit of it, only one arch 
should be employed. The rise of the arch is limited by the form 
of the roadway, and by the height of the banks of the river 
(See Art. 401.) The rise of the roadway should not exceed one 
in 24 feet, but, as the framing settles about one in 72, the roadway 
should be framed to rise one in 18, that it may be one in 24 after 
settling. The commencement of the arch at the abutments — the 
spring, as it is termed, should not be below high-water mark : 
and the bridge should be placed at right angles with the course of 
the current. 

406. — The best material for the abutments and piers of a 
bridge, is stone ; and, if possible, stone should be pr3cured for the 
purpose. The following rule is to determine the extent of the 
abutments, they being rectangular, and built with stone weighing 
120 lbs. to a cubic-foot. I^ule LXYII. — Multiply the square 
of the height of the abutment by 160, and divide this product by 
the weight of a square foot of the arch, and by the rise of the arch ; 
add unity to the quotient, and extract the square- root. Diminish 
the square-root by unity, and multiply the root, so diminished, by 



298 AMEPvICAN HOUSE-CARPENTER. 

half the span of the arch, and by the weiglit of a square-foot of 
the arch. Divide the last product by 120 times the height of the 
abutment, and the quotient will be the thickness of the abutment. 
Example. — Let the height of the abutment from the base to the 
springing of the arch be 20 feet, half the span 100 feet, the Aveight 
of a square foot of the arcli, including the greatest possible load 
upon it, 300 pounds, and the rise of the arch 18 feet — what should 
be its thickness ? The square of the height of the abutment, 
400, multiplied by 160, gives 64,000, and 300 by 18, gives 5400 ; 
64,000, divided by 5400, gives a quotient of 11*852, one added to 
this makes 12'S52, the square-root of which is 3'6 ; this, less one, 
is 2'6 ; this, multiplied by 100, gives 260, and this again by 300, 
gives 78,000 ; this, divided by 120 times the height of the abut- 
ment, 2400, gives 32 feet 6 inches, the thickness required. 

The dimensions of a pier will be found by the same rule. 
For, although the thrust of an arch may be balanced by an ad- 
joining arch, when the bridge is finished, and while it remains 
uninjured ; yet, during the erection, and in the event of one arch 
being destroyed, the pier should be capable of sustaining the en- 
tire thrust of the other. 

407. — Piers are sometimes constructed of timber, their princi- 
pal strength depending on piles driven into the earth, but such 
piers should never be adopted where it is possible to avoid them ; 
for, being alternately wet and dry, they decay much socner than 
the upper parts of the bridge. Spruce and elm are considered 
good for piles. Where the height from the bottom of the 
river to the roadway is great, it is a good plan to cut them off at 
a little below low- water mark, cap them with a horizontal tie. 
and upon this erect the posts for the support of the roadway. 
This method cuts off the part that is continually wet from that 
which is only occasionally so, and thus affords an opportunity for 
replacing the upper part. The pieces which are immersed will 
last a great length of time, especially when of elm ; for it is a 
well-established fact, that timber is less durable when subKJCt to 



FRAMING. 



299 



ilternt te dryness and moisture, than when it is either continually 
wet or continually dry. It has been ascertained that the piles 
ander London bridge, after having been driven about 600 years, 
urere not materially decayed. These piles are chiefly of elm, and 
vhoUy immersed. 




Fig. 25T. 



408. — Centres for stone bridges. Fig. 257 is a design fov a 
centre for a stone bridge where intermediate supports, as piles 
driven into the bed of the river, are practicable. Its timbers are 
so distributed as to sustain the weight of the arch-stones as they 
are being laid, without destroying the original form of the centre ; 
and also to prevent its destruction or settlement, should any of the 
piles be swept away. The most usual error in badly-constructed 
centres is, that the timbers are disposed so as to cause the framing 
to rise at the crown, during the laying of the arch-stones up the 
sides. To remedy this evil, some have loaded the crown with 
heavy stones ; but a centre properly constructed will need no 
such precaution. 

Experiments have shown that an arch-stone does not press 
upon the centring, until its bed is inclined to the horizon at an 
angle of from 30 to 45 degrees, according to the hardness of the 
stone, and whether it is laid in mortar or not. For general pur- 
poses, the point at which the pressure commences, may be con- 
sidered to be at that joint which forms an angle of 32 degrees 
with the horizon. At this pi^uit, the pressure is inconsiderable, 



300 



AMERICAN HOUSE-CAilPENTEB. 



but gradually increases towards the crown. The following 
table gives the j[)OTtion of the weight of the arch stones that 
presses upon the framing at the various angles of inclination 
formed by the bed of the stone with the horizon. The press- 
ure perpendicular to the curve is equal to the weight of the 
arch stone multiplied by the decimal 

•0, when the angle of inclination is 32 degrees. 



•04, 


(C « 


<( « 


" 


" 34 


•08 


il l( 


« (( 


« 


" 36 


•12 


« « 


« « 


« 


" 38 


•17 


(( <( 


« « 


« 


" 40 


•21 


« (( 


« (( 


« 


" 42 


•25 


(( It 


« « 


« 


" 44 


•29 


« « 


(( <t 


<( 


" 46 


•33 


« (( 


« (( 


« 


" 48 


•37 


« « 


" « 


" 


" 50 


•4 


It « 


(( « 


« 


" 62 


•44 


« « 


« (( 


« 


" 64 


•48 


(« « 


<t (( 


« 


« 56 


•52 


« (( 


(( « 


« 


" 68 


•64 


« i< 


« (( 


« 


" 60 




From this it is seen that at the inclination of 44 degrees the 
pressure equals one-quarter the weight of the stone ; at 57 de- 
grees, half the weight ; and when a vertical 
line, as a 5, {.Fig. 258,) passing through the 
centre of gravity of the arch-stone, does not 
fall within its bed, c d, the pressure may be 
considered equal to the whole weight of the 
stone. This will be the case at about 60 de- 
grees, when the depth of the stone is double its breadth. The 
direction of these pressures is considered in a line with the ra- 
dius of the curve. The weight upon a centre being known, 
the pressure may be estimated and the timber calculated ac- 
cordingly. But it must be remembered that the whole weight 
is never placed upon the framing at once — as see!ns to have 



Fig. 268. 



FRAMING. 



801 



been the idea had in view by the designers of some centres. 
In building the arch, it thould be commenced at each buttress 
at the same time, (as is generally the case,) and each side 
should progress equally towards the crown. In designing the 
framing, the effect produced by each successive layer of stone 
should be considered. The pressure of the stones npon one 
side should, by the arrangement of the struts, be counterpoised 
by that of the stones upon the other side. 

409. — Over a river whose stream is rapid, or where it is ne- 
cessary to preserve an uninterrupted passage for the purposes 
of navigation, the centre must be constructed without interme- 
diate supports, and without a continued horizontal tie at the 




base ; such a centre is shown at Fig. 259. In laying the stones 
from the base up to a and c, the pieces, h d and h d^ act as ties 
to prevent any rising at h. After this, while the stones are 
being laid from a and from c to 5, they act as struts : the piece, 
fg^ is added for additional security. Upon this plan, with 
some variation to suit circumstances, centres may be con- 
structed for any span usual in stone-bridge building. 

410. — In bridge centres, the principal timbers should abut, 
and not be intercepted by a suspension or radial piece between. 
These should be in halves, notched on each side and bolted. 
The timbers should intersect as little as possible, for the more 



302 AMERICAN HOIJSE-CAEPENTEK. 

joints the greater is tlie settling ; and halving them together is 
a bad practice, as it destroys nearly one-half the strength of 
the timber. Ties should be introduced across, especially where 
many timbers meet ; and as the centre is to serve but a tem- 
porary purpose, the whole should be designed with a view to 
employ the timber afterwards for other uses. For this reason, 
all unnecessary cutting should be avoided. 

411. — Centres should be sufficiently strong to preserve a 
staunch and steady form during the whole process of building; 
for any shaking or trembling will have a tendency to prevent 
the mortar or cement from setting. For this purpose, also, 
the centre should be lowered a trifle immediately after the 
key-stone is laid, in order that the stones may take their bear- 
ing before the mortar is set; otherwise the joints will open on 
the under side. The trusses, in centring, are placed at the 
distance of from 4 to 6 feet apart, according to their strength 
and the weight of the arch. Between every two trusses, diago- 
nal braces should be introduced to prevent lateral motion. 

412. — In order that the centre may be easily lowered, the 
frames, or trusses, should be placed upon wedge-formed sills ; 
as is shown at d^ {-Fig. 259.) These are contrived so as to ad- 
mit of the settling of the frame by driving the wedge, d^ with 
a maul, or, in large centres, with a piece of timber mounted as 
a battering-ram. The operation of lowering a centre should 
be very slowly performed, in order that the parts of the arch 
may take their bearing uniformly. The wedge pieces, instead 
of being placed parallel with the truss, are sometimes made 
sufficiently long and laid through the arch, in a direction at 
right angles to that shown at Fig. 259. This method obviates 
the necessity of stationing men beneath the arch during the 
process of lowering ; and was originally adopted with success 
Boon after the occurrence of an accident, in lowering a centre, 
by which nine men were killed. 

413. — To give some idea of the manner of estimating tlie pres 



FRAMING. 303 

enres, in order to select timber of the proper scantling, calculate 
{Art. 408) the pressure of the arch-stones from i to 5, (-Z^^'<7- 259,) 
and suppose half this pressure concentrated at a, and acting in 
the direction af. Then, by the parallelogram of forces, {.Art, 
258,) the strain in the several pieces composing the frame, 
hda, may be computed. Again, calculate the pressure of that 
portion of the arch included between a and <?, and consider 
half of it collected at h, and acting in a vertical direction ; 
then, by the parallelogram of forces, the pressure on the beams, 
h d and h d^ may be found. Add the pressure of that portion 
ol the arch which is included between i and h to half the 
weight of the centre, and consider this amount concentrated 
at d^ and acting in a vertical direction ; then, by constructing 
the parallelogram of forces, the pressure upon dj may be as- 
certained. 

414. — The strains having been obtained, the dimensions of 
the several pieces in the frames had and h g d^ may be found 
by computation, as directed in the case of roof trusses, from 
Arts. 8Y5 to 380. The tie-beams 1) d^h d., if made of sufficient 
size to resist the compressive strain acting upon them from the 
load at 5, will be more than large enough to resist the tensile 
strain upon them during the laying of the first part of the 
arch-stones below a and c. 

415. — In the construction of arches, the voussoirs, or arch- 
stones, are so shaped that the joints between them are perpen- 
dicular to the curve of the arch, or to its tangent at the point 
at which the joint intersects the curve. In a circular arch, the 
joints tend toward the centre of the circle : in an elliptical 
arch, the joints may be found by the following process : 




304: 



AMEEICAN HOUSE-CAEPENTER. 



416. — To find the direction of the joints for cm elliptical 
arch, A joint being wanted at «, {Fig. 260,) draw lines from 
that point to the foci,/" and/*/ bisect the ^iU^Qyfaf with the 
line, ah ; then a l will be the direction of the joint. 



/ //'''^c 


S 

f 


^L/^ 


. \ 


fa 


\ 



Fig. 26t 



417. — To find the direction of the joints for a jpardholic arch 
A joint being wanted at «^, {Fig. 261,) draw a e^ at right angles! 
to the axis, eg ; make c g equal to c e^ and join a and g j draw 
a A, at right angles to ag ; then a h will be the direction of 
the joint. The direction of the joint from 1) is found in the 
same manner. The lines, a g and hf are tangents to the curve 
at those points respectively; and any number of joints in the 
curve may be obtained, by first ascertaining the tangents, and 
then drawing lines at right angles to them. 







JOINTS. 




1 • 


f 


4 


^T- 









Fig. 268. 

418. — Fig. 262 shows a simple and quite strong method of 
lengthening a tie-beam ; but the strength consists wholly in 
the bolts, and in the friction of the parts produced by screwing 
the pieces firmly together. Should the timber shrink to even 
a small degree, the strength would depend altogether on the 
bolts. It would be made much stronger by indenting the 
pieces together ; as at the upper edge of the tie-beam in Fig. 
263 ; or by placing keys in the joints, as at the lower edge in 



FRAMING. 305 

the same figure. This process, however, weakens the beam in 
proportion to the depth of the indents. 



f" 



CJ- 



^=f 



Fig. 263. 



419. — JF'ig. 264 shows a method of scarfing, or splicing, a 
tie-beam without bolts. The keys are to be of well-seasoned, 

< '^=^ — en — p i 

Fig. 264. 

Hard wood, and, if possible, very cross-grained. The addition 
of bolts would make this a very strong splice, or even white- 
oak pins would add materially to its strength. 



Fig. 265. 

420. — Mg. 265 shows about as strong a splice, perhaps, as 
can well be made. It is to be recommended for its simplicity ; 
as, on account of there being no oblique joints in it, it can be 
readily and accurately executed. A complicated joint is the 
worst that can be adopted ; still, some have proposed joints 
that seem to have little else besides complication to recom 
mend them. 

421. — In proportioning the parts of these scarfs^ the depths 
of all the indents taken together should be equal to one-third 
of the depth of the beam. In oak, ash or elm, the whole 
length of the scarf should be six times the depth, or thickness, 
of the beam, when there are no bolts ; but, if bolts instead of 
indents are used, then three times the breadth ; and, when both 
methods are combined, twice the depth of the beam. The 

39 



306 AMEKICAN HOIJSE-CAEPENTER. 

length of the scarf in pine and similar soft woods, depending 
wholly on indents, should be about 12 times the thickness, or 
depth, of the beam ; when depending wholly on bolts, 6 times 
the breadth ; and, when both methods are combined, 4 times 
the depth. 



Fig. 266. 

422. — Sometimes beams have to be pieced that are required 
to resist cross strains — such as a girder, or the tie-beam of a 
roof when supporting the ceiling. In snch beams, the fibres 
of the wood in the upper part are compressed ; and therefore 
a simple butt joint at that place, (as in Fig. 266,) is far prefer- 
able to any other. In such case, an oblique joint is the very 
worst. The under side of the beam being in a state of tension, 
it must be indented or bolted, or both ; and an iron plate un- 
der the heads of the bolts, gives a great addition of strength. 

Scarfing requires accuracy and care, as all the indents should 
bear equally ; otherwise, one being strained more than another, 
there would be a tendency to splinter off the parts. Hence 
the simplest form that will attain the object, is by far the best. 
In all beams that are compressed endwise, abutting joints, 
formed at right angles to the direction of their length, are at 
once the simplest and the best. For a temporary purpose. Fig. 
262 would do very well ; it would be improved, however, by 
having a piece bolted on all four sides. Fig. 263, and indeed 
each of the others, since they have no oblique joints, would 
resist compression well. 

423. — In framing one beam into another for bearing pui 
poses, such as a floor-beam into a trinmier, the best place to 
make the mortice in the trimmer i's in the neutral line, {ArU 
317, 318,) which is in the middle of its depth. Some have 
thought that, as the fibres of the upper edge are compressed, a 



FRAMING. 307 

mortice might be made there, and the tenon driveL in tight 
enough to make the parts as capable of resisting the compres- 
sion, as thej would be without it ; and thej have therefore 
concluded that plan to be the best. This could not be the case, 
even if the tenon would not shrink; for a joint between two 
pieces cannot possibly be made to resist compression, so well 
as a solid piece without joints. The proper place, therefore, 
for the mortice, is at the middle of the depth of the beam ; but 
the best place for the tenon, in the floor-beam, is at its bottom 
edge. For the nearer this is placed to the upper edge, the 
greater is the liability for it to splinter off; if the joint is 



1 



Fig. 281. 

formed, therefore, as at Mg. 267, it will combine all the ad- 
vantages that can be obtained. Double tenons are objection- 
able, because the piece framed into is needlessly weakened, 
and the tenons are seldom so accurately made as to bear 
equally. For this reason, unless the tusk at a in the figure fits 
exactly, so as to bear equally with the tenon, it had better be 
omitted. And in sawing the shoulders, care should be taken 
not to saw into the tenon in the least, as it would wound the 
beam in the place least able to bear it. 

4,24:. — Thus it will be seen that framing weakens both pieces, 
more or less. It should, therefore, be avoided as much as pos- 
sible ; and where it is practicable one piece should rest upon 
the other, rather than be framed into it. This remark applies' 
to the bearing of floor-beams on a girder, to the purlins and 
jack-rafters of a roof, &c. 

425. — In a framed truss for a roof, bridge, partition, &c., 
the joints should be so constructed as to direct the pressures 



308 



AMERICAN HOUSE- CAKPENTEK. 



througli the axes of the several pieces, and also to avoid ever^ 
tendency of the parts to slide. To attain this object, the abut- 




Fig. 268. 



Fig. 270. 



ting surface on the end of a strut should be at right angles to 
the direction of the pressure ; as at the joint shown in Fig. 
268 for the foot of a rafter, (see Art. 277,) in Fig. 269 for the 
head of a rafter, and in Fig. 270 for the foot of a strut or 
brace. The joint at Fig. 268 is not cut completely across the 
tie-beam, but a narrow lip is left standing in the middle, and 
a corresponding indent is made in the rafter, to prevent the 
parts from separating sideways. The abutting surface should 
be made as large as the attainment of other necessary objects 
will admit. The iron strap is added to prevent the rafter slid- 
ing out, should the end of the tie-beam, by decay or otherwise, 
splinter off. In making the joint shown at Fig. 269, it should 
be left a little open at a., so as to bring the parts to a fair bear- 
ing at the settling of the truss, which must necessarily take 
place from the shrinking of the king-post and other parts. If 
the joint is made fair at first, when the truss settles it will cause 
it to open at the under side of the rafter, thus throwing the 
whole pressure upon the sharp edge at a. This will cause an 
indentation in the king-post, by which the truss will be made 
to settle further ; and this pressure not being in the axis of the 
rafter, it will be greatly increased, thereby rendering the rafter 
liable to split and break. 

426. — If the rafters and struts were made to abut end to 
end, as in Figs. 271, 272 and 273, and the king or queen post 
notched on in halves and bolted, the ill effects of shrinking 



fUAMI^ifG. 



309 



would be avoided. This metliod has been practised with suc- 
cess, in some of the most celebrated bridges and roofs in Eu- 



^'■^ 




Fig. 271. 



Fig. 272. 



Fig. 273. 



rope ; and, were its use adopted in this countiy, the unseemly 
sight of a hogged ridge would seldom be met with. A plate 
of cast iron between the abutting surfaces will equalize the 
pressure. 





Fig. 274 



Fig. 275. 



427. — I^ig. 274 is a proper joint for a collar-beam in a small 
roof: the principle shown here should characterize all tie- 
joints. The dovetail joint, although extensively practised in 
the above and similar cases, is the very w^orst that can be em- 
ployed. The shrinking of the timber, if only to a small de- 
gree, permits the tie to withdraw — as is shown at I^ig. 275. 
The dotted line shows the position of the tie after it has 
shrunk. 

428. — Locust and white-oak pins are great additions to the 
strength of a joint. In many cases they w^ould supply the 
place of iron bolts ; and, on account of their small cost, they 
should be used in preference wherever the strength of iron ig 



310 AMERICAN HOUSE-CAKPENTER. 

not requisite. In small framing, good cut nails are of great 
service at the joints ; but they should not be trusted to bear 
any considerable pressure, as they are apt to be brittle. Iron 
straps are seldom necessary, as all the joinings in carpentry 
may be made without them. They can be used to advantage, 
however, at the foot of suspending-pieces, and for the rafter at 
the end of the tie-beam. In roofs for ordinary purposes, the 
iron straps for suspending-pieces may be as follows : When the 
longest unsupported part of the tie-beam is 

10 feet, the strap may be 1 inch wide by j\ thick. 
15 " " " li " " i " 

20 " " " 2 " " i " 

In fastening a strap, its hold on the suspending-piece will be 
much increased by turning its ends into the wood. Iron straps 
should be protected from rust ; for thin plates of iron decay 
very soon, especially when exposed to dampness. For this 
purpose, as soon as the strap is made, let it be heated to about 
a blue heat, and, while it is hot, pour over its entire surface 
raw linseed oil, or rub it with beeswax. Either of these will 
give it a coating which dampness will not penetrate. 



IRON GEBDEKS. 




Fig. 27& Fig. 277. 

4^9.— Fig. 276 represents the front view, and Fig. 277 the 
cross section at middle, of a cast iron girder of proper form 
f)r sustaining a weight equally diffused over its length. The 
curve is that of a parabola : generally an arc of a circle is 



FKAMING. 311 

used, and is near enough. Beams of this form are much used 
to sustain brick walls of buildings ; the brickwork resting upon 
the bottom flange, and laid, not arching, but horizontal. In 
the cross section, the bottom flange is made to contain in area 
four times as much as the top flange. The strength will be in 
proportion to the area of the bottom flange, and to the height 
or depth. Hence, to obtain the greatest strength from a given 
amount of material, it is requisite to make the upright part, or 
the blade, rather thin ; yet, in order to prevent injurious strains 
in the casting while it is cooling, the parts should be nearly 
equal in thickness. The thickness of the three parts — blade, 
top flange and bottom flange, may be made in proportion as 5, 
6 and 8. For a beam of this form, the weight equally dif- 
fused over it equals 

w = 9000 ^. (199.) 

The depth equals 

d = -l^. . (200.) 

9000 ta ^ ' 

The area of the bottom flange equals 

where w equals the weight in pounds equally diflfused over the 
length ; d, the depth, or height in inches of the cross section 
at middle ; a, the area of the bottom flange in inches ; /, the 
length of the beam in feet, in the clear between the bearings ; 
and t, a decimal in proportion to unity as the safe weight is to 
the breaking weight. This is usually from 0*2 to 0*3, or from 
one-fifth to one-third, at discretion. 

430. — Beams of this form, laid in series, are much used in 
sustaining brick arches turned over vaults and other fire-proof 
rooms, forming a roof to the vault or room, and a floor above ; 
the arches springing from the flanges, one on either side of the 
beam, as in Mg. 278. 



312 



AMERICAN HOUSE-CAitPENTER. 




his:. Wi 



(202.) 



For this use the depth of cross section at middle equals 

._ c/r 

~ 9000 t a 
The area of the bottom flange equals 



a = 



(203.) 



9000 t d' 

where the symbols signify as before, and c equals the distance 
apart from centres in feet at which the beams are placed, and 
f the weight per superficial foot, in pounds, including the 
weight of the material of which the floor is constructed. . 



Practical Rules and Examples. 

431. — For a single girder the dimensions may be found by 
the following rule, ((200) and (201) :) 

Bule LXYIII. — Divide the weight in pounds equally dif- 
fused over the length of the girder by a decimal in proportion 
to unity as the safe weight is to the breaking weight, multiply 
the quotient by the length in feet, and divide tlie product by 
9000. Then this quotient, divided by the depth of the beam 
at middle, will give the area of the bottom flange ; or, if di- 
vided by the area of the bottom flange, will give the depth — • 
the area and depth both in inches. 

Example. — Let the weight equally diff*used over a girder 
equal 60000 pounds ; the decimal that is in proportion to unity 
as the safe weight is to be to the breaking weight, equal 0*3 



FEAi^ima. 313 

the length in the clear of the bearings equal 20 feet. Then 
60000 divided bj 0-3 equals 200000, and this by 20 equals 
4000000 ; this divided by 9000 equals 44:4|. E"ow if the depth 
Is fixed, say at 20 inches, then M4:|-, divided by 20, equals 22|, 
equals the area of the bottom flange in inches. But if the 
area is given, say 24 inches, then to find the depth, divide 
4M|- by 24, and the quotient, 18*5, equals the depth in inches; 
and such a girder may be made with a bottom flange of 2 by 
12 inches, top flange, (equal to \ of bottom flange,) IJ by 4 
inches, and the blade li inches thick. 

432. — For a series of girders or iron beams, the dimensions 
may be found by the following rule ; (202) and (203). 

Rule LXIX. — Divide the weight per superficial foot, in 
pounds, by a decimal in proportion to unity as the safe weight 
is to the breaking weight, and multiply the quotient by the 
square of the length of the beams and by the distance apart at 
which the beams are placed from centres, both in feet, and di- 
vide the product by 9000. Then this quotient, divided by the 
depth of the beams at middle, will give the area of the bottom 
flange ; or, if divided by the area of the bottom flange, will 
give the depth of the beam — the depth and area both in inches. 

Example. — Let the weight per superflcial foot resting upon 
an arched floor be 200 pounds, and the weight of the -arches, 
concrete, &c., equal 100 pounds, total 300 pounds per superfi- 
cial foot. Let the proportion of the breaking weight to be 
trusted on the beams equal 0-3, the length of the beams in the 
clear of the bearings equal 12 feet, and the distance apart from 
centres at which they are placed equal 4 feet. Then 300 di- 
vided by 0-3 equals 1000 ; this multiplied by 144 (the square 
of 12), equals 144000, and this by 4, equals 576000 ; this di- 
vided by 9000, equals 64. Now if the depth is fixed, and at 
8 inches, then 64 divided by 8 equals 8, equals the area of the 
bottom flange. But if the area of the bottom flange is fixed, 
and at 6 inches, then 64, divided by 6, equals 10|, the depth 

40 



314 AMERICAN HOUSE-CARPENTEK. 

required. Such a beam may be made with the bottom flange 
1 by 6 inches, the top flange, (equal to one-quarter of the bot- 
tom flange,) f by 2 inches, and the blade f inch thick. 

433. — The kind of girder shown at Fig, 280, (a cast iron 
arch with a wrought iron tie rod,) is extensively used as a sup- 
port upon which to build brick walls where the space below is 
required to be free. The objections to its use are, the dispro- 
portion between the material and the strains, and the enhanced 
cost over the cast iron girder foi-med as in Figs. 276 and 277. 
The material in the cast arch, {Fig. 280,) is greatly in excess 
over the amount needed to resist efi'ectually the compressive 
strains induced by the load through the axis of the arch, while 
the wrought metal in the tie is usually much less than is re- 
quired to resist the horizontal thrust of the arch ; absolute fail- 
ure being prevented, partly by the weight of the walls resting 
on the haunches, and partly by the presence of adjoining 
buildings, their walls acting as buttresses to the arch. Some 
instances have occurred where the tie has parted. 

Where this arched girder is used it is customary to lay the 
first courses of brick in the form of an arch. This brick arch 
of itself is quite sufficient to sustain the compressive strain, 
and, were there proper resistance to the horizontal thrust pro- 
vided, the brick arch would entirely supersede the necessity 
for the girder. Indeed, the instances are not rare where con- 
structions of this nature have proved quite satisfactory, the 
horizontal thrust of the arch being sustained by a tie rod 
secured to a pair of cast iron heel plates, as in Fig. 279. The 




Fig. 279. 



FE AMINO. 315 

brick arcli, in tliis case, being built upon a wooden centre, 
which was afterwards removed. 

The diameter of the rod required for an arch of this kind is 
eoual to 



D 



^/mh (^^^•) 



where w equals the weight in pounds equally diffused over the 
arch ; 5, the length of tlie rod, clear of the heel plates, in feet ; 
and /i, the height at the middle, or rise, of the arc, in inches ; 
j9, the diameter, being also in inches. 

When the diameter found by formula (204) is impracticably 
large, this difficulty may be overcome by dividing the metal 
into two rods. In the bow-string girder, {Fig. 280,) two rods 
cannot be used with advantage, because of the difSculty in 
adjusting their lengths so as to ensure to each an equal amount 
of the strain. But in the case of the brick arch, the two heel 
plates being disconnected, any discrepancy of length in the 
rods is adjusted simply by the pressure of the arch acting on 
the plates. When there are to be two rods, the diameter of 
each rod equals 

J)=J ^\, (205.) 

Practical Bute and Examjple. 

434. — To obtain the diameter of wrought iron tie-rods foi 
heel plates, as in Fig. 279, proceed by this rule. 

Rule LXX. — Multiply the weight in pounds equally distri- 
buted over the arch by the length of the tie-rod in feet, cleai 
of the heel plates, and divide the product by the height of the 
arc in inches, (that is, the height at the middle, from the axis 
of the tie-rod to the centre of the depth of the brick arch,) then, 
if there is to be but one tie-rod, divide the quotient by 3000 ; 



316 



AMEKIOAN HOUSE-CAKPENTEE. 



but if two, tlien divide by 6000, and the square root of the 
quotient, in either case, will be the required diameter. 

Example. — The weight to be supported on a brick arch, 
equally distributed, is 24000 pounds ; the length of the tie-rod, 
clear of the heel plates, is 10 feet ; and the height, or rise, of 
the arc is 10 inches. Now by the rule, 24000 X 10 = 240000. 
This divided by 10, equals 24000. Upon the presumption that 
one tie-rod only will be needed, divide by 3000, and the ^ .u- 
tient is 8, the square root of which is 2'82 inches. This is 
"ather large, therefore there had better be two rods. In this 
'.ase the quotient, 24000, divided by 6000, equals 4, the square 
^-oot of which is 2, the diameter required. The arch should, 
therefore, have two rods of 2 inches diameter. Two rods are 
preferable to one. The iron is stronger per inch in small rods 
than in large ones, and the rules require no more metal in the 
two rods than in the one. 




Fig. 280. 



435. — The Bow-string Girder^ as per Fig. 280, has little to 
recommend it, (see Art. 433,) yet because it has by some been 
much used, it is well to show the rules that govern its strength, 
if only for the benefit of those who are willing to be governed 
by reason rather than precedent. To resist the horizontal 
thrust of the cast arch, the diameter of the rod must equal 

(204) 

/_ws_ 
V 3000 A' 

But the cast iron arch has a certa'n amount of strength to re 



FRAMTNG. S17 

sigt cross strains : this strength must be considered. Upon the 
presumption that the cross section of the cast arch at the mid- 
dle is of the most favorable form, as in J^ig. 277, or at least 
that it have a bottom flange, (although the most of those cast 
are without it), the strength of the cast arch to resist cross 
strains is shown by formula (199), when ?, its length, is changed 
to s, its span. The weight in pounds equally diffused over thr* 
arch will then equal 

9000 t a d 

W = ; 

S 

This is the weight borne by the cast arch acting simply as a 
beam. Deducting this weight from the whole weight, the re- 
mainder is the weight to be sustained by the rod. Calling the 
whole weight w, then 

9000 tad ws — 9000 tad = 'W 

w — — 

s s 

Therefore, fiom (204), the diameter equals 



- = 7: 



3000 A 



U v) s - 900 1 1 



3000 A 
WS" 9000 t a d 



(206.) 



3000 h ' 

where D equals the diameter of the rod in inches ; w^ the 
weight in pounds equally diffused over the arch ; 5, the span 
of the arch in feet ; A, the rise or height of tlie arc at middle, 
in inches ; d^ the height or depth of the cross section of the 
cast arch in inches ; a^ the area of the bottom flange of the 
cross section of the cast arch in inches ; and ^, a decimal in 
proportion to unity as the safe weight is to be to the breaking 
weight. 

The rule in words at length, is 

Rule LXXI. — Multiply the decimal in proportion to unity 



318 AMERICAN HOUSE-CARPENTER. 

as the safe weight is to be to the breaking weight, bj 000(? 
times the depth of the cross section of the cast arch at middle, 
and by the area of the bottom flange of said section, both in 
inches, and deduct the product from the weight in pounds 
equally diffused over the arch multiplied by the span in feet, 
and divide the remainder by 3000 times the height of the arc 
in inches, measured from the axis of the tie-rod to the centre 
of the depth of the cast arch at middle, and the square root 
of the quotient will be the diameter of the rod in inches. 

Example. — The rear wall of a building is of brick, and is 40 
feet high, and 21 feet wide in the clear between the piers of 
the story below. Allowing for the voids for windows, this 
wall will weigh about 63000 pounds ; and it is proposed to 
support it by a bow-string girder, of which the cross section at 
middle of the cast arch is 8 inches deep, and has a bottom 
flange containing 12 inches area. Tlie rise of the curve or arc 
is 24 inches. What must be the diameter of the rod, the por- 
tion of the breaking weight of the cast arch, considered safe 
to trust, being three-tenths or 0*3 ? By the rule, 0-3 x 9000 
X 8 X 12 = 259200 ; then 63000 x 21 - 259200 = 1063800. 
This remainder divided by (3000 x 24 =) 72000, the quotient 
equals 14'775 ; the square root of which, 3*84, or nearly 3| 
inches, is the required diameter. 

This size, though impracticably large, is as small as a due 
regard for safety will permit ; yet it is not unusual to find the 
rods in girders intended for as heavy a load as in this exam- 
ple, only 2i and 2J- inches ! "Were it possible to attach the 
rod so as not to injure its strength in the process of shrinking 
it in — putting it to its place hot, and depending on the con- 
traction of the metal in cooling to bring it to a proper bearing 
— and were it possible to have the bearings so true as to induce 
the strain through the axis of the rod, and not along its side^ 
{Art. 308,) then a less diameter than that given by the rule 
would suffice. But while these contingencies remain, the rule 



FRAMING. 319 

cannot safely be reduced, for, in the rule, the value of T, for 
wrought iron, (Table III., Art. 308,) is taken at nearly 600C 
pounds, a point rather high in consideration of the size of the 
rod and the injuries, before stated, to which it is subjected. 
In cases where a girder wholly of cast iron {I^ig. 276) is not 
preferred, it were better to build a brick arch resting on heel 
plates, {Mg, 279,) in which the metal required to resist the 
thrust ma}^ be divided into two rods instead of one, thus render- 
ing the size more practical, and at the same time avoiding the 
injuries to which rods in arch girders are subjected. The heel- 
plate arch is also to be preferred to the cast arch on the score 
of economy ; inasmuch as the brick which is substituted for 
the east arch will cost less than iron. For example, suppose 
the cross section of the iron arch to be thus : the blade or up- 
right part 8 by 1-| inches, the top flange 12 by IJ inches, and 
the bottom flange 6 by If inches. At these dimensions, the 
area of the cross section will equal 12 + 15 + 10|- = 37^ 
inches. A bar of cast iron, one foot long and one inch square, 
will weigh 3*2 pounds; therefore, S7i X 3*2 = 120 pounds, 
equals the weight of the cast arch per lineal foot. The price 
of castings per pound, as also the price of brickwork per 
cubic foot, of course will depend upon the locality and the 
state of the market at the time, but for a comparison they may 
be stated, the one at three and a half cents per pound, and the 
other at thirty cents per cubic foot. At these prices the cast 
arcli will cost 120 x 3^ = $4: 20 per lineal foot; while the 
brick arch — 12 inches high and 12 inches thick — will cost 30 
cents per lineal foot. The difi^erence is $3 90. This amount 
is not all to be credited to the account of the brick arch 
Proper allowance is to be made for the cost of the heel plates, 
and of the wooden centre ; also for the cost of a small addi- 
tion to the size of the tie rods, which is required to sustain the 
strain otherwise borne by the cast arch in its resistance to a 
cross strain {Art. 435). Deducting the cost of these items, 



320 AMERICAN HOUSE-CAEPENTER. 

the difference in favor of the brick arch will be about $3 pei 
foot. This, on a girder 25 feet long, amounts to $75. The 
difference in all cases will not equal this, but will be sufficiontl/ 
great to be worth saving. 



SECTION v.— DOORS, WINDOWS, &c. 



DOORS. 

436. — Among the several architectural arrangements of an edi- 
fice, the door is by no means the least in importance ; and, if pro- 
perly constructed, it is not only an article of use, but also of or- 
nament, adding materially to the regularity and elegance of the 
apartments. The dimensions and style of finish of a door, should 
be in accordance with the size and style of the building, or the 
apartment for which it is designed. As regards the utility of 
doors, the principal door to a public building should be of suffi- 
cient width to admit of a free passage for a crowd of people ; 
while that of a private apartment will be wide enough, if it per- 
mit one person to pass without being incommoded. Experience 
has determined that the least width allowable for this is 2 feet 8 
inches ; although doors leading to inferior and unimportant rooms 
may, if circumstances require it, be as narrow as 2 feet 6 inches ; 
and doors for closets, where an entrance is seldom required, may 
be but 2 feet wide. The width of the principal door to a public 
building may be from 6 to 12 feet, according to the size of the 
building ; and the width of doors for a dwelling may be from 2 
feet 8 inches, to 3 feet 6 inches. If the importance of an apart- 
ment in a dwelling be such as to require a door of greater width 

4:1 






322 AMERICAN HOUSE-CAJ^PENTER. 

than 3 feet 6 inches, the opening should be closed with two 
doors, or a door in two folds ; generally, in such cases, where the 
opening is from 5 to 8 feet, folding or sliding doors are adopted. 
As to the height of a door, it should in no case be less than about 
6 feet 3 inches ; and generally not less than 6 feet 8 inches. 

437. — The proportion between the width and height of single 
doors, for a dwelling, should be as 2 is to 5 ; and, for entrance- 
doors to public buildings, as 1 is to 2. If the width is given and 
the height required of a door for a dwelling, multiply the width 
by 5, and divide the product Dy 2 ; but, if the height is given and 
the width required, divide by 5, and multiply by 2. Where two 
or more doors of different widths show in the same room, it is 
well to proportion the dimensions of the more important by the 
above rule, and make the narrower doors of the same height as 
the wider ones ; as all the doors in a suit of apartments, except 
the folding or sliding doors, have the best appearance when of 
one height. The proportions for folding or sliding doors should 
be such that the width may be equal to J of the height ; yet this 
rule needs some qualification : for, if the width of the opening 
be greater than one-half the width of the room, there will not be 
a sufficient space left for opening the doors ; also, the height 
should be about one-tenth greater than that of the adjacent single 
doors. 

438. — Where doors have but two panels in width, let the stiles 
and muntins be each -j of the width ; or, whatever number of 
panels there may be, let the united widths of the stiles and the 
muntins, or the whole width of the solid, be equal to -f of the width 
of the door. Thus : in a door, 35 inches wide, containing two 
panels in width, the stiles should be 5 inches wide ; and in a door, 
3 feet 6 inches wide, the stiles should be 6 inches. If a door, 3 
feet 6 inches wide, is to have 3 panels in width, the stiles and 
muntins should be each 4.^ inches wide, each panel being 8 inches. 
The bottom rail and the lock rail ought to be each equal' in 
width to yV of the height of the door ; and the top rail, and all 



J 



-" A 



DOORS, WINEOV.S, &€. M'Jo 

Others, of the same width as the stiles. The moulding on the 
panel should bo equal in width to i of the width of the stiJp 




Fig. 281. 



439. — Pig, 281 shows an approved method of trimming doors : 
a is the door stud ; 6, the lath and plaster ; c, the ground ; d^ the 
jamb ; e, the stop ; /and g, architrave casings ; and A, the door 
stile. It is customary in ordinary work to form the stop for the 
door by rebating the jamb. But, when the door is thick and 
heavy, a better plan is to nail on a piece as at e in the figure. 
This piece can be fitted to the door, and put on after the door is 
hung ; so, should the door be a trifle windings this will correct 
the evil, and the door be made to shut solid. 

440. — Fis^. 282 is an elevation of a door and trimminars suita- 
ble for the best rooms of a dwelling. (For trimmings generally, 
see Sect. III.) The number of panels into which a door should 
be divided, is adjusted at pleasure ; yet the present style of finish- 
ing requires, that the number be as small as a proper regard for 
strength will admit. In some of our best dwellings, doors have 
been made having only two upright panels. A few years expe- 
ience, however, has proved that the omission of the lock rail 
is at the expense of the strength and durability of the door ; a 
four-panel door, therefore, is the best that can be made. 

441. — The doors of a dwelling should all be hung so as to open 
into the principal rooms ; and, in general, no door should be hung 
to open into the hall, or passage. As to the proper edge of the 
door on which to affix the hinges, no general rule can be assigned 



324 



AMERICAl^' HOUSE-CAEPEXTER. 




Fig. 288. 



WINDOWS. 



442. — A window should be of such dimensions, and in such 
a position, as to admit a suiSciency of light to that part of the 
apartment for which it is designed. "No definite rule for the size 



DOORS, WINDOWS, &C. 32a 

can well be given, that will answer in all cases ; yet, as an ap- 
proximation, the following has been used for general purposes, 
Multiply together the length and the breadth in feet of the apart- 
ment to be lighted, and the product by the height in feet ; then 
the square-root of this product will show the required number ol 
square feet of glass. 

443. — To ascertain the dimensions of window frames, add 4| 
inches to the width of the glass for their width, and 6^ inches to 
the height of the glass for their height. These give the dimen- 
sions, in the clear, of ordinary frames for 12-light windows ; the 
height being taken at the inside edge of the sill. In a brick wall, 
the width ot the opening is 8 inches more than the width of the 
glass — 4^ for the stiles of the sash, and 3^ for hanging stiles — 
and the height between the stone sill and lintel is about 10 g inches 
more than the height of the glass, it being varied according to the 
thickness of the sill of the frame. 

444. — In hanging inside shutters to fold into boxes, it is ne- 
cessary to have the box shutter about one inch wider than the 
flap, in order that the flap may not interfere when both are folded 
mto the box. The usual margin shown between the face of the 
shutter when folded into the box and the quirk of the stop bead, 
or edge of the casing, is half an inch ; and, in the usual method 
of letting the whole of the thickness of the butt hinge into the 
todge of the box shutter, it is necessary to make allowance for the 
thi'ow of the hinge. This may, in general, be estimated at i of 
an inch at each hinging ; which being added to the margin, the 
entire width of the shutters will be l-g inches more than the width 
of the frame in the clear. Then, to ascertain the width of the 
box shutter, add I5 inches to the width of the frame in the clear, 
between the pulley stiles ; divide this product by 4, and add 
half an inch to the quotient ; and the last product will be the re- 
quired width. For example, suppose the window to have 3 
lights in width, 11 inches each. Then, 3 times 11 is 33, and 4^ 
added for the wood of the sash, gives 37;^ 37f and 1;^ is 39 



326 



AMERICAN HOUSE CARPENTER. 



and 39. divided by 4, gives 9| ; to which add half an inch, and' 
the result will be 10| inches, the width required for the box shutter. 
4:4:0. — In disposing and proportioning windows for the walls of 
a building, the rules of architectural taste require that they be of 
different heights in different stories, but of the same width. The 
windows of the upper stories should all range perpendicularly 
over those of the first, or principal, story ; and they should be 
disposed so as to exhibit a balance of parts throughout the front 
of the building. To aid in this, it is always proper to pl;tce the 
front door in the middle of the front of the building ; and, where 
the size of the house will admit of it, this plan should be adopted. 
(See the latter part oi^ Art. 224.) The proportion that the height 
should bear to the width, may be, in accordance with general 
usage, as follows : 

The height of basement windows, 1^ of the width. 
" " principal-story " 2j " 

" " second-story " 1| " 

" " third-story " 1| * 

" " fourth-story "1^ " 

" " attic-story " the same as the width. 

But, in determining the height of the windows for the several 
stories, it is necessary to take into consideration the height of the 
story in which the window is to be placed. For, in addition to 
the height from the floor, which is generally required to be from 
28 to 30 inches, room is wanted above the head of the window 
for the window-trimming and the cornice of the room, besides 
some respectable space which there ought to be between these. 

446. — Doors and windows are usually square- headed, or termi- 
nate in a horizontal line at top. These require no special direc- 
tions for their trimmings. But circular-beaded doors and win- 
dows are more difficult of execution, and require some attention. 
If the jambs of a door or window be placed at right angles to the 
face of the wall, the edges of the soffit, or surface of the head, 
would be straight, and its length be found by getting the 



DOOKS, WINDOWS, &C. 



321 



stretch-out of tlie circle, {An. 92;) but, when the jaiibs are 
placed obliquely to the face of the wall, occasioned by the de- 
mand for light in an oblique direction, the form of the soffit 
wdll be obtained by the following article : and, when the face 
of the wall is circular, as in the succeeding one. 

/ 




Fig. 283. 



447. — To find the form of the soffit for circular luindow 
heads, when the light is received in an oblique directiofi. Let 
abed, {Fig. 283,) be the ground-plan of a given windoAv, ande/ 
a, a vertical section taken at right angles to the face of the jambs. 
From a, through e, draw ag, at right angles to a b ; obtain the 
stretch-out of ef a, and make e g equal to it ; divide e g and e 
f a, each into a like number of equal parts, and drop perpen- 
diculars from the points of division in each ; from the points of 
intersection, 1, 2, 3, &c., in the line, a d, draw horizontal lines to 
meet corresponding perpendiculars from eg; then those points 
of intersection will give the curve line, d g, which will be the 
one required for the edge of the soffit. The other edge, c A, is 
found in the same manner. 

448. — To find the form of the soffit for circular windoin- 
heads, when the face of the wall is curved. Let abed, {Fig. 
284,) be the ground-plan of a given window, and e/ «, a vertical 
section of the head taken at right angles to the face of the jambs 



328 



AMERICAN HOUSE-CARPENTFR. 















e 


A 


^ 






^ 


\ 






7"^ 






— 1 — 






— 








< 


a 


?-- 
















-'6 




^^"^ 












"""cJ 




^~- 









1 


__ -"^ 




^ I 


2 


/■ d 


/ 


z 


3 


A 


^"■^^ 












— 






l- 




^ 




^"«v,,,_^ 




1 










1 


^,,0'^ 






^^^^^^ — 


(=r^ 




—^z^ 


-==- 


'-"^ 





Fig. 284. c 

Proceed as in the foregoing article to obtain the line, d g; thei: 
that will be the curve required for the edge of the soffit; the 
other edge being found in the same manner. 

If the given vertical section be taken in a line with the face of 
the wall, instead of at right angles to the face of the jambs, place 
it upon the line, c b, (Fig. 283 ;) and, having drawn ordinates at 
right angles to c b, transfer them to ef a ; in this way, a section 
at right angles to the jambs can be obtained. 



SECTION VL— STAIRS. 



4:4:9. — The stairs is that mechanical arrangement in a build- 
ing by which access is obtained from one story to another. Theii 
position, form and finish, when determined with discriminating 
taste, add greatly to the comfort and elegance of a structure. As 
regards their position, the first object should be to have them near 
the middle of the building, in order that an equally easy access 
may be obtained from all the rooms and passages. Next in im- 
portance is light; to obtain which they would seem to be best 
situated near an outer wall, in which windows might be construc- 
ted for the purpose ; yet a sky-light, or opening in the roof, would 
not only provide light, and so secure a central position for the 
stairs, but may be made, also, to assist materially as an ornament 
to the building, and, what is of more importance, aiford an op- 
portunity for better ventilation. 

450. — It would seem that the length of the raking side of the 
^itch-hoard^ or the distance from the top of one riser to the top ot 
the next, should be about the same in all cases ; for, whether stairs 
be intended for large buildings or for small, for public or for pri- 
vate, the accommodation of men of the same stature is to be con- 
sulted in every instance. But it is evident that, with the same 
effort, a longer step can be taken on level than on rising ground 

42 



330 



AMERICAN HOUSE-CARPKJSTKR. 



and that, although the tread and rise cannot be proportioned 
merely in accordance with the style and importance of the build- 
ing, yet this may be done according to the angle at which the 
flight rises. If it is required to ascend gradually and easy, the 
length from the top of one rise to that of another, or the hypothe 
nuse of the pitch-board, may be long ; but, if the flight is steep 
the length must be shorter. Upon this data the following piobleii 
is constructed. 




451. — To proportion the rise and tread to one another. 
Make the line, a b, {Fig. 285,) equal to 24 inches ; from b, ereci 
b c, at right angles to a b, and make b c equal to 12 inches ; join a 
and c, and the triangle, a b c, will form a scale upon which to 
graduate the sides of the pitch-board. For example, suppose a 
very easy stairs is required, and the tread is fixed at 14 inches. 
Place it from b to/, and from/; draw/^, at right angles to a b ; 
then the length of f g will be found to be 5 inches, which is a 
proper rise for 14 inches tread, and the angle, f b g, will show 
ihe degree of inclination at which the flight will ascend. But, in 
a majority of instances, the height of a story is fixed, while the 
length of tread, or the space that the stairs occupy on the lovvei 
floor, is optional. The height of a story being determined, the 
height of each rise will of course depend upon the number into 
which the whole height is divided ; the angle of ascent being more 
easy if the number be great, than if it be smaller. By dividing 



STAIRS. 331 

the whole height oi a story into a certain number of rises, sup- 
pose the length of each is found to be 6 inches. Place this length 
from b to A, and draw h i, parallel to a b ; then h i, or bj will be 
the proper tread for that rise, and J b i will show the angle of as- 
cent. On the other hand, if the angle of ascent be given, as a 
b I, {b I being 10|^ inches, the proper length of ru?i for a step- 
ladder,) drop the perpendicular, I k, from I to k ; then I k b will 
be the proper proportion for the sides of a pitch-board for that 
run. 

452. — The angle of ascent will vary according to circum- 
stances. The following treads will determine about the right in- 
clination for the different classes of buildings specified. 

In public edifices, tread about 14 inches. 

In first-class dwellings " 12^ " 

In second-class " " 11 ^* 

In third-class " and cottages " 9 " 

Step-ladders to ascend to scuttles, &c., should have from 10 tc 
11 inches run on the rake of the string. (See notes at Art. 103.' 
453. — The length of the steps is regulated according to the ex- 
tent and importance of the building in which they are placed, 
varying from 3 to 12 feet, and sometimes longer. Where two per- 
sons are expected to pass each other conveniently, the shortest 
length that will admit of it is 3 feet ; still, in crowded cities where 
land is so valuable, the space allowed for passages being very 
small, they are frequently executed at 2^ feet. 

454. — To find the dimensions of the pitch-board. The first 
thing in commencing to build a stairs, is to make the^i^cA-board ; 
this is done in the following manner. Obtain very acciuately, in 
feet and inches, the perpendicular height of the story in which 
the stairs are to be placed. This must be taken from the top ol 
the floor in the lower story to the top of the floor in the upper 
story. Then, to obtain the number of rises, the height in inches 
tJms obtained must be divided by 5, 6, 7, 8, or 9, according to the 
quality and style of the building in which the stairs are to ba 



8^2 AMERICAN HOUSE-CARPENTER. 

built. For instance, suppose the building to be a fijst-class 
dwelling, and the height ascertained is 13 feet 4 inches, or 160 
inches. The proper rise for a stairs in a house of this class is 
about 6 inches. Then, 160 divided by 6, gives 26| inches.. This 
being nearer 27 than 26, the number of risers, should be 27. 
Then divide the height, 160 inches, by 27, and the quotient will 
give the height of one rise. On performing this operation, the 
quotient will be found to be 5 inches, |- and j\ of an inch. 

Then, if the space for the extension of the stairs is not limited, 
the tread can be found as at Art. 451. But, if the contrary is the 
case, the whole distance given for the treads must be divided by 
the number of treads required. On account of the upper floor 
forming a step for the last riser, the number of treads is always 
one less than the number of risers. Having obtained this 
rise and tread, the pitch-board may be made in the follow- 
ing manner. Upon a piece of well-seasoned board about f of an 
inch thick, having one edge jointed straight and square, lay the 
corner of a carpenters'-square, as shown at Fig. 286. Make a b 




Fig 286. 



equal to the rise, and b c equal to the tread ; mark along those 
edges with a knife, and cut it out by the marks, making the edges 
perfectly square. The grain of the wood must run in the direction 
indicated in the figure, because, if it shrinks a trifle, the rise and 
the tread will be equally affected by it. When a pitch-board is 
first made, the dimensions of the rise and tread should be pre- 
served in figures, in order that, should the first shrink, a second 
could be made. 
455. — 7^0 lay out the string. The space roquired for timbei 




333 



t'la. 287. 



and plastering under the steps, is about 5 inches for ordinary 
stairs ; set a gauge, therefore, at 5 inches, and run it on the lowei 
edge of the plank, as a b, {Fig. 287.) Commencing at one end, 
lay the longest side of the pitch-board against the gauge-mark, a 
6, as at c, and draw by the edges the lines for the first rise and 
tread: then place it successively as at d, e and/, until the re- 
quired number of risers shall be laid down. 



UH 



Fig. 288. 

4.56. — Fig. 288 represents a section of a step and riser, joined 
after the most approved method. In this, a represents the end of 
a block about 2 inches long, two of which are glued in the corner 
in the length of the step. The cove at b is planed up square, 
glued in, and stuck after the glue is set. 



PLATFORM STAIRS. 

457. — A platform stairs ascends from one story to another in 
two or more flights, having platforms between for resting and 
to cnange their direction. This kind of stairs is the most easily 
constructed, and is therefore the most common. The cjlin- 



nu 



AMERICAN HOUgE-CARPENTER, 




der is generally of small diameter, in most cases about 6 inches. 
It may be worked out of one solid piece, but a better way is lo 
glue together three pieces, as in Fig. 2(S9 ; in which the pieces, 
a, h and c, compose the cylinder, and d and e represent parts of 
the strings. The strings, after being ghied to the cylinder, are 
secured with screws. The joining at o and o is the most proper 
for that kind of joint. 

4:58. — To obtain the form of the lower edge of the cylinder. 
Find the stretch-out, d e, {Fig. 290,) of the face of the cylinder 
ah c^ according to Art. 92 ; from d and e, draw d f and e g, at 
right angles to 6^ e ; draw h g, parallel to d e, and make hf and 
g i, each equal to one rise; from i and/, draw ij and/ A:, paral- 
lel to h g ; place the tread of the pitch-board at these last lines, 
and draw by the lower edge the lines, k h and i I ; parallel to 
tnese, draw m n and o jo, at the requisite distance for the dimen- 
sions of the string ; from 5, the centre of the plan, draw ^9 a. 
parallel to df; divide h q and q g, each into 2 equal parts, as at 
V and w ; from v and iv, draw v n and lu o, parallel tofd; join n 
and 0, cutting q s in r ; then the angles, u n r and r o t^ being 
eased off according to Art. 89, will give the proper curve for the 
bottom edge of the cylinder. A centre may be found upon which 
to describe these curves thus : from ?i, draw u x^ at right angles 
to m n ; Irom r, draw r .r, at right angles Xo no ; then x will be 
the centre for the curve, u r. The centre for the ?,Ufve r t^ \^ 
foiind in the same manner. 



STAIRS. 



^83 




Fig. 290. 

459. — To find the position for the balusters. Place the 
centre of the first baluster, ih. Fig. 291,) | its diameter from the 
face of the riser, c c?, and k its diameter from the end of the step, 
e d ; and place tlie centre of the other baluster, a, half the tread 
from the centre of the first. The centre of the rail must be placed 
over the centre of the balusters. Their usual length is 2 feet 
5 inches, and 2 feet 9 inches, for tht short and the long balusters 
respectively. 



e^^3i 



Tig. 291. 



336 



AMERICAN HOUSE-CARPENTER. 



^\ 


I 




./3 


'n 


T e 


a 


^ 


S 








a 


^ 


5 




U 





Fig. 292. 



460. — To find the face-mould for a round hand-rail to plat 
form stairs. Case 1. — When the cylinder is small. In Fig, 
292, j and e represent a vertical section of the last two steps of the 
first flight, and d and i the first two steps of the second flight, of 
a platform stairs, the line, e f being the platform ; and a b c is 
the plan of a line passing through the centre of the rail around 
the cylinder. Through i and d, draw i k, and through J and e, 
draw 7 k ; from k, draw k Z, parallel to / e ; from b, draw b m, 
parallel to^ d; from Z, draw I r, parallel to k j ; from n^ draw 71 
t, at right angles to^ k ; on the line, o b. make t equal to n t ; 
join c and t : on the line, j c, {Fi^. 293 ) make e c equal to e n at 
Fig. 292 ; from c, draw c t, at right angles toj c, and make c t 



STAIRS. 337 




equal to c ^ at Fig. 292 ; through t, draw/? Z, parallel to J c, and 
make 1 1 equal to ^ Z at Fig. 292 ; join I and c, and complete the 
parallelogram, e c Is; find the points, o, o, o, according to Art. 
118 ; upon e, o, o, o, and/, successively, with a radius equal to 
half the width of the rail, describe the circles shown in the figure ; 
then a curve traced on both sides of these circles and just touch- 
ing them, will give the proper form for the mould. The joint at 
I is drawn at right angles to c I. 

461. — Elucidation of the foregoing method. This excellent 
plan for obtaining the face-moulds for the hand-rail pf a platform 
stairs, has never before been published. It was communicated to 
me by an eminent stair-builder of this city : and having seen 
rails put up from it, I am enabled to give it my unqualified re- 
commendation. In order to have it fully understood, I have in- 
troduced Fig. 294 ; in which the cylinder, for this purpose, is 
made rectangular instead of circular. The figure gives a per- 
spective view of a part of the upper and of the lower flights, and 
a part of the platform about the cylinder. The heavy lines, i m^ 
m c and c /, show the direction of the rail, and are supposed to 
pass through the centre of it. When the rake of the second 
flight is the same as that of the first, whicl. is here and is gene- 
rally the case, the face-mould for the lower twist will, when re- 
versed, do for the upper flight: that part of the rail, therefore, 
which passes from e to c and from c to /, is all that will need ex* 
planation. 

Suppose, then, that the parallelogram, e a o c^ represent a plane 
lying perpendicularly over e ab f being inclined in the direction, 
e c, and level in the direction, c o ; suppose this plane, e a o c, 

43 



338 



AMERICAN HOUSE-CARPENTER. 




Fig. 294. 



be revolved on e c as an axis, in the manner indicated by the arcs, 
n and a x^ until it coincides with the plane, e r t c ; the line, a 
9. will then be represented by the line, a; n ; then add the paral- 
lelogram, xrt n, and the triangle, ctl, deducting the triangle, ers, 
and the edges of the plane, e s I c, inclined in the direction, ec, and 
also in the direction, c I, will lie perpendicularly over the plane, e 
a bf. From this we gather that the line, c o, being at right angles tfl 



STAIRS. 



339 



e c, must, in order to reach the point, Z, be lengthened the distance, 
n t^ and the right angle, e ct^hQ made obtuse by the addition to 
it of the angle, t c I. By reference to Fig. 292, it will be seen 
that this lengthening is performed by forming the right-angled 
triangle, c o t^ corresponding to the triangle, c o ^, in Fig. 294. 
The line, c t^ is then transferred to Fig. 293, and placed at right 
angles to e c; this angle, e c t^ being increased by adding the an- 
gle, t c Ij corresponding to t c I, Fig. 294, the point, /, is reached, 
and the proper position and length of the lines, e c and c I ob- 
tained. To obtain the face-mould for a rail over a cylindrica' 
well-hole, the same process is necessary to be followed until the 
the length and position of these lines are found ; then, by forming 
the parallelogram, eels, and describing a quarter of an ellipse 
therein, the proper form will be given. 




Fig 295. 



4:62.— Case 2 — When the cyliyider is large. Pis'. 295 



re- 



uo 



AMERICAN HOUSE-CARPENTER. 



presents a plan and a vertical section of a line p.issing through the 
centre of the rail as before. From b, draw b k, parallel io cd ; ex- 
tend the lines, i d and J e, until they meet kbink and/; from w, 
draw n I, parallel to ob ; through Z, draw I t, parallel tojk, from 
kj draw k t, at right angles ioj k ; on the line, o b, make o t equal 
to k t. Make e c, {Fig. 296.) equal to e k at Fig: 295 ; from c, 




Fig. 296. 

draw c t, at right angles to e c, and equal to c ^ at Pig, 295 • from 
t, draw t p, parallel to c e, and make 1 1 equal to tl oX Fig. 295 ; 
complete the parallelogram, eels, and find the points, o, o, o, as 
before ; then describe the circles and complete the mould as in 
Fig. 293 The difference between this and Case 1 is, that the 
line, c t, instead of being raised and thrown out, is lowered and 
drawn in. (See note at page 381.) 




FU. 2a7. 



463. — Case 3. — Where the rake meets the level. In Fig 



STAIRS. 



341 



297, abcis the plan of a line passing through the centre it tne 
rail around the cyUnder as before, and j and 6? is a vertical sectior. 
of two steps starting from the floor, h g. Bisect e hind, and 
through d, draw df, paraUel to h g ; bisect /n in I, and Irom /, 
draw I t, paraUel to nj; from n, draw n t, at right angles tojn^ 
on the line, o b, make o t equal to n t. Then, to obtain a mould 
for the twist going up the flight, proceed as at Fig. 293 ; making 
e c in that figure equal to e n in Fig. 297, and the other lines ot 
a length and position such as is indicated by the letters of reference 
in each figure. To obtain the mould for the level rail, extend h 
0, (Fig. 297,) to i ; make o i equal to / /, and join i and c ; maU-q 
c i, (Fig. 298,) equal to c i at Fig. 297 ; through c, draw c 4. at 



d C 

Fig. 298. 



right angles to ci ; make d c equal to dfai Fig. 297, and com 
plete the parallelogram, o d c i; then proceed as in the previous 
cases to find the mould. 

464. — All the moulds obtained by the preceding examples have 
been for round rails. For these, the mould may be applied to 
a plank of the same thickness as the rail is intended to be, and 
the plank sawed square through, the joints being cut square from 
the face of the plank. A twist thus cut and truly rounded will 
hang in a proper position over the plan, and present a perfect and 
graceful wreath. 

465. — To bore for the balusters of a round rail before ro md- 
ing it. Make the angle, o c t, {Fig. 299,) equal to the angle, o 
c ^, at Fig. 292 ; upon c, describe a circle with a radius equal to 
half the thickness of the rail ; draw the tangent, b d, parallel to 
t c, and complete the rectangle, e b df having sides tangical to 
the circle; from c, draw c «, at right angles to oc; then, b d 
being the bottom of the ra"l, set a gauge from 6 to a, and run it 
the whole length of the stuff; in boring, place the centre ot th.i 



M9. 



AMERICAN HOUSE-CARPENTER. 
b 




bit in the gauge-mark at a, and bore in the directionj a c. To do 
this easily, make chucks as represented in the figure, the bottom 
edge, g A, being parallel to o c, and having a place sawed out, as 
ej, to receive the rail. These being nailed to the bench, the rail 
will be held steadily in its proper place for boring vertically. 
The distance apart that the balusters require to be, on the under 
side of the rail, is one-half the length of the rake-side of the 
pitch-board. 




Fig. £00. 



STAIRS, 



34?, 



466. — To obtain, by the foregoing principles, the face-mould 
for the twists of a moulded rail upon platform stairs In Fig, 
300, a 6 c is the plan of a line passing through the centre of 
the rail around the cylinder as before, and the lines above 
It are a vertical section of steps, risers and platform, w^ith 
the lines for the rail obtained as in Fig. 292. Set half the width 
of the rail from b to f and from b to r, and from / and r, draw/ 
e and r d, parallel to c a At Fig. 301, the centre lines of the 

s d__n_J 




rai'., k c and c n, are obtained as in the previous examples. Make 
c i and c j, each equal to c i at Fig. 300, and draw the hnes, i m 
and ;' g, parallel lo c k ; make n e and n d equal to n e and n d dX 
Fig. 300, and draw d o and e I, parallel to n c ; also, through k, 
draw s g, parallel to n c ; then, in the parallelograms, m s d o and 
g s e I, find the elliptic curves, d m and e g, according to Art. 
118, and they will define the curves. The fine, d e, being drawn 
through n parallel to k c, defines the joint, which is to be cut 
through the plank vertically. If the rail crosses the platform rather 
steep, a butt joint will be preferable, to obtain wlch see Art. 498. 




344 AMERICAN HOUSE-CARPENTE.R. 

467. — To apply the mould to the plank. The mould obtained 
according to the last article must be applied to both sides of the 
plank, as shown at Fig. 302. Before applying the mould, the 
edge, ef, must be bevilled according to the angle, c ^ ^, at Fis( 
800 ; if the rail is to be canted up. the edge must be bevilled at 
an obtuse angle with the upper face ; but if it is to be canted 
down.) the angle that the edge makes with the upper face mnstbe 
acute. From the spring of the curve, a, and the end, c, draw 
vertical lines across the edge of the plank by applying the pitch- 
board, a b c ; then, in applying the mould to the other side, place 
the points, a and c, at 6 and/; and, after marking around it, saw 
the rail out vertically. After the rail is sawed out, the bottom 
and the top surfaces must be squared from the sides. 

468. — To ascertain the thickness of stuff I'equired for the 
twists. The thickness of stuff required for the tAvists of a round 
rail, as before observed, is the same as that for the straight ; but 
for a moulded rail, the stuff for the twists must be thicker than 
that for the straight. In Fig. 300, draw a section of the rail be- 
tween the lines, d r and e f and as close to the line, d e, as possi- 
ble ; at the lower corner of the section, draw ^ A, parallel to d e; 
then the distance that these lines are apart, will be the thickness 
required for the twists of a moulded rail. 

The foregoing method of finding moulds for rails is applicable 
to all stairs which have continued rails around cylinders, and are 
without winders. 

WINDING STAIRS. 

469. — Winding stairs have steps tapering narrower at one end 
than at the other. In some stairs, there are steps of parallel width 
incorporated with tapering steps ; the former are then called^yer^ 
and the latter tnnders. 

470. — To describe a regular geometrical winding stairs. 
In Fig. 303, abed represents the inner surface of the wall en 
closing the space allotted to the stairs, a e the length of the steps, 
and efgh the cylinder, or face of the front string. The line, 



STAIRS. 



345 




U^ 



a e, IS given as the face of the first riser, and the point, jj for the 
limi: of the last. Make e i equal to 18 inches, and upon o, with 
i for radius, describe the arc, ij; obtain the number of risers 
and of treads required to ascend to the floor atj, according to Art. 
454, and divide the arc, ij, into the same number of equal parts 
as there are to be treads ; through the points of division, 1, 2, 3, 
»fcc., and from the wall-string to the front-string, draw lines tend- 
ing to the centre, o ; then these lines will represent the face ot 
each riser, and determine the form and width of the steps. Allow 
the necessary projection for the nosing beyond a e, which should 
be equal to the thickness of the step, and then ael k will be the 
dimensions for each step. Make a pitch-board for the wall-string 
having a k for the tread, and the rise as previously ascertained ; 
with this, lay out on a thicknessed plank the several risers and 
treads, as at Fig. 287, gauging from the upper edge of the strirg 
for the line at which to set the pitch-board. 

Upon the back of the string, with a i^ inch dado plane, maki 

44 



346 



AMERICAN HOUSE-CARPENTER 



a succession of grooves 1^^ inches apart, and parallel with the 
lines for the risers on the face. These grooves must be f ut along 
the whole length of the plank, and deep enough to admit of the 
plank's bending around the curve, abed. Then construct a 
drum, or cylinder, of any common kind of stuff, and made to fit 
a curve having a radius the thickness of the string less than o a ; 
upon this the string must be bent, and the grooves filled with strips 
of wood, called keys, which must be very nicely fitted and glued 
in. After it has dried, a board thin enough to bend around on the 
outside of the string, must be glued on from one end to the other 
and nailed with clout nails. In doing this, be careful not to nail 
into any place where a riser or step is to enter on the face. 

After the string has been on the drum a sufficient time for the 
glue to set, take it off", and cut the mortices for the steps and 
risers on the face at the lines previously made ; which may be 
done by boring with a centre-bit half through the string, and 
nicely chiseling to the line. The drum need not be made so 
large as the whole space occupied by the stairs, but merely large 
enough to receive one piece of the wall-string at once — for it 
is evident that more than one will be required. The front string 
may be constructed in the same manner ; taking e I instead of a 
k for the tread of the pitch-board, dadoing it with a smaller dado 
plane, and bending it on a drum of the proper size. 




Fig. 804. 

471. — To find the shape and position of the timhei^s fieces' 
sary to support a winding stairs. The dotted lines in Fig. 
303 show the proper position of the timbers as regards the plan : 
the shape of each is obtained as follows. In Fig. 304, the line, 
1 a, is equal to a riser, less the thickness of the floor, and the 
lines, 2 m, 3 /i, 4 o, 5 /? and 6 q, are each equal to one riser. The 



STAIRS. 



347 



line, a 2, is equal to a m in Fig. 303, the line, m 3 torn n in that 
figure, &c. In drawing this figure, commence at a, and make 
the lines, a 1 and a 2, of the length above specified, and dra^^ 
them at right angles to each other ; draw 2 m, at right angles to 
a 2, and m 3, at right angles to m 2, and make 2 m and m 3 of 
the lengths as above specified ; and so proceed to the end. Then, 
through the points, 1, 2, 3, 4, 5 and 6, trace the line, 16; upon 
the points, 1, 2, 3, 4, &c., with the size of the timber for radius, 
describe arcs as shown in the figure, and by these the lower line 
may be traced parallel to the upper. This will give the proper 
shape for the timber, a b, in Fig. 303 ; and that of the others may 
be found in the same manner. In ordinary cases, the shape of 
one face of the timber will be sufficient, for a good workman 
can easily hew it to its proper level by that ; but where great 
accuracy is desirable, a pattern for the other side may be found 
in the same manner as for the first. 

472. — To find the falling-mould for the rail of a winding 
stairs. In Fig. 305, a ch. represents the plan of a rail around 
half the cylinder, A the cap of the newel, and 1, 2, 3, <fec., the 
face of the risers in the order they ascend. Find the stretch-out, 
c/, of a c 6, according to Art. 92; from o, through the point of 
the mitre at the newel-cap, draw o s ; obtain on the tangent, e d, 
the position of the points, s and h\* as at t and/' ; from e tf^ and 
/; draw e x^t u^f^ g^ and /A, all at right angles Xo e d ; make e 
g equal to one rise and/^^^ equal to 12, as this line is drawn 
from the 12th riser ; from g^ through g^^ draw^ i; make g x equal 
to about three-fourths of a rise, (the top of the newel, x^ should 
be 3 J feet from the floor ;) draw x w, at right angles to e x^ and 
ease off the angle at w ; at a distance equal to the thickness of 

* In the above, the references, a^, 6^, &c., are introduced for the first time. During the 
lime taken to refer to the figure, the memory of the form of these may pass from the mind, 
While that of the sound alone remains ; they m£ y then be mistaken for a 2, 6 2, &c. This 
«in be avoided in reading by giving them a so.nd corresponding to their meaning, which 
is second a second b, <fec. or a second, b second. 



3J-S 



AMERICAN HOUSE-CARPENTER. 




Fiff. 305. 



tho rail, draw v w y, parallel to x ui; from the centre of the plan, 
0, draw o I, at right angles to e d ; bisect h n in p, and through 
J), at right angles to^ i, draw a line for the joint ; in the same 
manner, draw the joint at k ; then x y will be the falling-mould 
for tliat part of the rail which extends from 5 to 6 on the plan. 

473. — To find tl -e face-mould for the railof axoinding-stairs. 
From the extremities of the joints in the falling-mould, as k^ z 
and y, {Fig. 305,) draw k a'^, z Jy^ and y </, at right angles to e d ; 
make b e^ equal to / d. Then, to obtain the direction of the 
joint, a^ c\ 01 b' d y proceed as at Fig 306, at which the pans are 



STAIRS. 



349 




Fig. 306. 



shown at half their full size. A is the plan of the rail, and B is 
the falling-mon'd ; in which k z is the direction of the butt-joint. 
From k^ draw k 5, parallel to / o, and k e, at right angles to k h : 
from 6, draw h /, tending to the centre of the plan, and from/, draw 
/ e, parallel io b k ; from /, through e draw I i, and from i, draw i 
d, parallel to e/; join d and 6, and rf 6 will be the proper direction 



3r.o 



AMERICAN HOUSE-CARPENTER. 



for the joint on the plan. The direction of the joint on the othei 
side, a c, can be found by transferring the distanajs, x h and o d 
io X a and o c. (See Art. 477.) 




Fig. SOT. 



Raving obtained the direction of the joint, make s r d b, {Fig. 
307,) equal to s r d' 6^ in Fig. 305 ; through r and d, draw t a , 
through A' and from d, draw t u and d e, at right angles to ^ o ; 
make t u and d e equal to t u and 6^ m, respectively, in Fig. 305 ; 
from 11^ through e, draw ii o ; through 6, from r, and from as man}^ 
other points in the line, t a, as is thought necessary, as/, h and ; 
draw the ordiiiates, r c,f g, h i^j k and a o ; from w, c, ^, r, A:, e 
and 0, draw the ordinates, w 1, c 2, ^ 3, i 4, A: 5, e 6 and o 7, at 
right angles to u o ; make u 1 equal to ^ 5, c 2 equal to r 2, ^ 3 
equal to/ 3, &c., and trace the curve, 1 7, through the points 
thus found ; find the curve, c e, in the same manner, by transfer- 
ring the distances between the line, t a, and the arc, r d ; join 1 
and c, also e and 7 ; then, 1 c e 7 will be the face-mould required 
for that part of the rail which is denoted by the letters, s r d^ h\ 
on the plan at Fig. 305. 

To ascertain the mould for the next quarter, make a cje, ( Fig 



BTAIRS. 




Fig. 308. 



308,) equal to a* c^ j e^ at Fig. 305 ; at any convenient height on 
the line, d i, in that figure, draw q i\ parallel to e d ; through c 
and/ [Fig. 308,) draw b d ; through a, and from J, draw h k and 
; 0, at right angles ^o b d ; make b k and j o equal to i^ k and q 
i, respectively, in Fig. 305 ; from k, through o, draw kf; and 
proceed as in the last figure to obtain the face-mould, A. 

474. — To ascertain the requisite thickness of stuff. Case 
1. — When the falling-mould is straight. Make o h and k m, 
(Fig. 308,) equal to i y at Fig. 305 ; draw h i and m w, parallel 
tob d ; through the corner farthest from kf, as n or i, draw n ^, 
parallel to kf ; then the distance between kf and n i will give 
the thickness required. 

475. — Case 2. — When the falling-mould is curved. In Fig. 
309, s r dbis equal to 5 r c?^ 6Mn Fig. 305. Make a c equal to the 
stretch-out of the arc, 5 b, according to Art. 92, and divide a c and 
s b. each into a like number of equal parts ; from a and c. and from 
each point of division in the line, ac, draw ak, el, 6cc., at right an- 
gles to fl c ; make a A: equal to t u in Fig. 305, and c ;equalto6'7ii 



B53 



AMERICAN HOUSE-CARPENTER. 




Fig. 319 



ill that figure, and complete the tailing-mould, k j^ every way eqiia 
to u m in Fig. 305 ; from the points of division in the arc, 56, draw 
lines radiating towards the centre of the circle, dividing the arc. 
r (^, in the same proportion as 5 6 is divided ; from d and 6, draw 
d t and h u^ at right angles to a c?, and from j and v^ draw J ii and % 
I/;, at right angles \oj c ; then x t uio will be a vertical projection 
of the joint, d b. Supposing every radiating iine across s r d b — 
corresponding to the vertical lines across A; j — to represent a joint, 
find their vertical projection, as at 1, 2, 3, 4, 5 and 6 ; through the 
corners of those parallelograms, trace the curvp lines shown in the 
figure ; then 6 u will be a helinet, or vertical projection, of 5 r tt b. 
To find the thickness of plank necessary to get out this part of 
the rail, draw the line, z t, touching the upper side of the helinet 
in two places : through the corner farthest projecting from that 
line, as w, draw y w, parallel to z t ; then the distance between 
those lines will be the proper thickness of stuff for this part of the 
rail. The same process is necessary to find the thickness of 
stuff in all cases in which the falling-mould is in any way curved. 
476. — To apply the face-mould to the plank. In Fig. 310, 
A represents the plank with its best side and edge in view, and 
B the same plank turned ap so as to bring in view the other side 



STAIRS. 



358 




Fig. 810. 



and the same edge, this being square from the face. Apply the 
tips of the mould at the edge of the plank, as at a and o, {A,) and 
mark out the shape of the twist ; from a and o, draw the lines, a 
h and o c, across the edge of the plank, the angles, e a h and e o 
c, corresponding with kfda.t Fig-. 308 ; turning the plank up as 
at B, apply the tips of the mould at b and c, and mark it out as 
shown in the figure. In sawing out the twist, the saw must be 
be moved in the direction, a b ; which direction will be perpen- 
dicular when the twist is held up in its proper position. 

In sawing by the face-mould, the sides of the rail are obtained ; 
the top and bottom, or the upper and the lower surfaces, are ob- 
tained by squaring from the sides, after having bent the falling- 
mould around the outer, or convex side, and marked by its edges. 
Marking across by the ends of the falling-mould will give the 
position of the butt-joint. 

4:77. — Elucidation of the process by which the direction of 
the butt-joint is obtained in Art. 473. Mr. Nicholson, in his 
Carpenter'^ s Guide, has given the joint a different direction to 
that here shown ; he radiates it towards the centre of the cylin- 
der. This is erroneous — as can be shown by the following 
operation : 

In Fig. 311, a rj iis theplan of a part of the rail about the 
joint, 5 w is the stretch-out of a i, and g p is the helinet, or ver- 
tical projection of the plan, a r j i, obtained according to Art 

45 



354 



AMERICAN HOirSE-CARPENTEH. 




Fig. 311 



i75. Bisect r t^ part of an ordinate from the centre of the fjlan, 
and through the middle, draw c &, at right angles io g v ; from 
h and c, draw c d and h e, at right angles to s u ; from d and e, 
draw lines radiating towards the centre of the plan : then d o 
and e m will be the direction of the joint on the plan, according to 
Nicholson, and c h its direction on the falling-mould. It will be 
admitted that all the lines on the upper or the lower side of the rail 
which radiate towards the centre of the cylinder, as c? o, e m or 
i J, are level ; for instance, the level line, w Vj on the top of the 



STAIRS. 



rail m the helinet, is a true representation of the radiating line, j I 
on the plan. The line, b h, therefore, on the top of the rail in 
the helinet, is a true representation of e m on the plan, and A: c on 
the bottom of the rail truly represents d o. From k, draw k /, 
parallel to c 6, and from h, draw hf, parallel to 6 c ; join I and 
b, also c andf; then c k I b will be a true representation of the 
end of the lower piece, B, and cfh b of the end of the uppei 
piece, A ; and/ k oxh I will show how much the joint is open on 
the inner, or concave side of the rail. 




356 



AMERICAN HOUSE-CARPENTER. 



To show that the process followed in Art. 473 is correct, let d o 
and e m, {Fig. 312,) be the direction of the butt-joint found as at 
Fig. 306. Now, to project, on the top of the rail in the helinet, a 
Jinethat does not radiate towards the centre of the cylinder, as J 
k, draw vertical lines from J and k to w and h, and join w and h , 
then it will be evident that wh is a true representation in the helinet 
of ) k on the plan, it being in the same plane as ; k^ and also in the 
same winding surface as w v. The liuC, I n, also, is a true repre- 
«entation on the bottom of the helinet of the line, J k, in the plan. 
The line of the joint, e m, therefore, is projected in the same way 
and truly by i 6 on the top of the helinet ; and the line, d o, by 
c <x on the bottom. Join a and i, and then it will be seen that 
the lines, c a, a i and i 6, exactly coincide with c b, the line of 
-he joint on the convex side of the rail ; thus proving the lower 
:.nd of the upper piece, A, and the upper end of the lower piece, 
B, to be in one and the same plane, and that the direction of the 
joint on the plan is the true one. By reference to Fig. 306 it will 
be seen that the line, I ^, corresponds to ^ i in Fig. 312 ; and 
that e k in that figure is a representation of/ 6, and i k oi db. 





Fig. 313. 



In getting out the twists, the joints, before the falling-mould is 



STAIRS 357 

applied, are cut perpendicularly, the flict moui^ being long enc>agh 
to include the oveiplus necessary for a butt-joint. Tl.e face-mould 
for A^ therefore, would have to extend to the line, i b ; and that foi 
B, to the line, y z. Being sawed vertically at first, a section of the 
joint at the end of the face-mould for A, would be represented in 
the helinet by h if g. To obtain the position of the line, h i, on 
the end of the twist, draw is, {Fig. 313,) at right angles to ij] 
and make i s equal to m e at Fig. 312 ; through 5. draw s g, pa- 
rallel to i /, and make s h equal to 5 6 at Fig. 312 ; join h and i ; 
make i/equal to i /at Fig. 312, and from /", draw /^, parallel to \ 
h ; then i b gf will be a perpendicular section of the rail over the 
line, e m, on the plan at Fig. 312, corresponding to i b gf in the 
helinet at that figure ; and when the rail is squared, the top, or 
back, must be trimmed off* to the line, i b, and the bottom to the 
line, fg. 

478. — To grade the fi^ont string of a stairs, having winders 
in a quarter-circle at the top of the flight connected with flyers 
at the bottom. In Fig. 314, a b represents the line of the facia 
along the floor of the upper story, bee the face of the cylinder, 
and c d the face of the front string. Make^ b equal to ^ of the 
diameter of the baluster, and draw the centre-line of the rail, /^, 
g h i and ij, parallel to a b, b e c and c d ; make g k and g I 
each equal to half the width of the rail, and through k and /, 
draw lines for the convex and the concave sides of the rail, parallel 
to the centre-line ; tangical to the convex side of the rail, and parallel 
to k ni, draw no; obtain the stretch-out, q r, of the semi-circle, k 
p m, according to Art. 92 ; extend ab to t, and k ni to s ; make c s 
equal to the length of the steps, and i it equal to 18 inches, and de- 
scribe the arcs, s t and u 6, parallel to m jt? ; from t, draw t w, tend- 
ing to the centre of the cylinder ; from 6, and on the line, 6 ux, run 
off" the regular tread, as at 5, 4, 3, 2, 1 and v ; make u x equal to 
half the arc, u 6, and make the point of division nearest to x, as 
?', the limit of the parallel steps, or flyers ; make r o equal to?n z ; 
from 0, draw a al at right angles to n o. and equal to one rise ; 



AMERICAN HOUSE-CARPENTEB, 




Fig. 814. 



from c^^ draw d s, parallel to 7t o, and equal to one tread ; from s 
through 0, draw 5 6^ 

Then from w, draw w c^, at right angles to n o, and set up, on 
tlie line, w c^, the same number of risers that the floor. A, is above 
the first winder, B, as at 1, 2, 3, 4, 5 and 6; through 5, (on the 
arc, 6 u,) draw (P e^, tending to the centre of the cylinder; from 
e\ draw e^/^, at right angles to n o, and through 5, (on the line, 



STAIRS. 359 

ID J^,) draw ^^/^ parallel tono ; through 6, (on the; line, w c\] 
and/^, draw the line, Iv b' ; make 6 coequal to half a rise, and 
from c^ and 6, draw c^ i^ and 6/, parallel ton o ; make /r r equal 
to A^/^; from i^, draw t^ k^, at right angles to i^ h'^, and from/~, 
draw/'^A;'^, at right angles to/ Vi%- upon P, with ^'^/^ for radius, 
describe the arc,/^ ^%• make b"^ P equal to b^f^, and ease off the 
angle at 6^ by the curve, /^ Zl In the figure, the curve is de- 
scribed from a centre, but in a full-size plan, this would be imprac- 
ticable ; the best way to ease the angle, therefore, would be with 
a tanged curve, according to Art. 89. Then from 1, 2, 3 and 4, 
(on the line, w c'^,) draw lines parallel to n o, meeting the curve in 
m"^, 71^, d^ and p^ ; from these points, draw lines at right angles to 
n 0, and meeting it in a;^, r"^, s'^ and f; from a;'^ and r*^, draw lines 
tending to u^, and meeting the convex side of the rail in y^ and 
z"^ ; make ni v^ equal to r /, and m iif equal to r f ; from y^, z'\ 
v"^, and w'^, through 4, 3, 2 and 1, draw lines meeting the line of 
the wall-string in a^, 6^, c^ and (f ; from e^, where the centre-line ot 
the rail crosses the line of the floor, draw e^/^, at right angles to w 
0, and from/^, through 6, draw/^ g"^ ; then the heavy lines,/^^^, 
e^ (f*, 7/"^ a^, 2;'^ 6^, v^ c^, -z^^ c?^ and 2; y, will be the lines for the risers, 
which, being extended to the line of the front string, b e c d, will 
give the dimensions of the winders, and the grading of the front 
string, as was required. 

479. — To obtain the falling-mould for the twists of the last- 
mentioned stairs. Make i^ ^^ and iVi^, (i^i^. 314,) each equal 
to half the thickness of the rail ; through h^ and g^^ draw h^ i^ 
andg^f, parallel to i^ s ; assuming k k^ and m, rr^ on the plan as 
the amount of straight to be got out with the twists, make n q 
equal to k k^, and r l^ equal to m m%- from n and /^, draw lines at 
right angles to n 0, meeting the top of the falling-mould in n^ and 
0^ ; from 0^, draw a line crossing the falling-mould at right angles 
to a chord of the curve, /^ P ; through the centre of the cylinder, 
draw iP 8, at right angles to no ; through 8, draw 7 9, tending to 
F; then^i^ 7 will be the falling-mould for the upper twist, and 7 
a* the falling-mould for the lower twist. 



360 



AMERICAN HOUSE-CARPENTER. 



480. — To ohtam the face-moulds. The moulds for the twists* 
of this stairs may be obtained as at Art. 473 ; but, as the faUing- 
mould in its course departs considerably from a straight line, it 
would, according to that method, require a very thick plank for 
the rail, and consequently cause a great waste of stuff. In order, 
therefore, to economize the material, the following method is to 
be preferred — in which it will be seen that the heights are taken 
in three places instead of two only, as is done in the previous 
method. 




Fig. 315. 



Case 1. — When the middle height is above a line joining 
the other tivo. Having found at Fig. 314 the direction of the 
joint, w s^ and p e, according to Art. 473, make k p e a, (Fig. 
315,) equal to k^ p^ e p in Fig. 314 ; join h and c, and from o, 
draw A, at right angles to 6 c ; obtain the stretch-out of d g, as 
df and at Fig. 314, place it from the axis of the cylinder, j)^ lo 
q^ ; from q^ in that figure, draw q^ r^, at right angles to n o ; also, 
at a convenient height on the line, n n^^ in that figure, and at 
right angles to that line, draw u^ v^ ; from b and c, in Fig. 315, 



STAIRS. SGI 

draw b j and c U n-t right angles to b c ; make bj equal to u^ n^ in 
Fig. 314, i A equal to w^ r" in that figure, and c I equal to 2;^ 9 ; 
from Z, through j', draw Im ; from ^, draw h n, parallel to c 6 y 
from 71, draw ti r, at right angles to b c, and join r and 5 ; through 
the lowest corner of the plan, as j)^ draw v e, parallel to b c ; from 
a, e, M, p, ^% ^5 and from as many other points as is tliought ne- 
cessary, draw ordinates to the base-line, v e, parallel to r s ; 
through A, draw w x^ at right angles to m I ; upon n, with r s for 
radius, describe an intersecting arc at x^ and join ?i and x ; from 
the points at which the ordinates from the plan meet the base- 
line, V e, draw ordinates to meet the line, m I, at right angles to v 
e ; and from the points of intersection on m Z, draw correspond- 
ing ordinates, parallel to n x ; make the ordinates which are pa- 
rallel to n X oi di length corresponding to those which are parallel 
to r 5, and through the points thus found, trace the face-mould 
as required. 

Case 2. — When the middle height is below a line joining 
the other tivo. The lower twist in Fig. 314 is of this nature. 
The face-mould for this is found at Fig. 316 in a manner similar 
to that at Fig. 315. The heights are all taken from the top of 
the falling-mould at Fig. 314 ; b j being equal to lo 6 in Fig. 314, 
i h equal to x^ y^ in that figure, and clXol^ ol Draw a Ime 
through J and Z, and from A, draw h n^ parallel to 6 c; from n., 
draw n r, at right angles to b c, and join r and s ; then r s will be 
the bevil for the lower ordinates. From A, draw h x, at right an- 
gles to j I ; upon 71, with r s for radius, describe an intersecting 
arc at x^ and join n and x ; then n x will be the bevil for the upper 
ordinates, upon which the face-mould is found as in Case 1. 

481. — Elucidation of the foregoing method. — This method 
of finding the face-moulds for the handrail ing of winding stairs, 
being founded on principles which govern cylindric sections, may 
be illustrated by the following figures. Fig. 317 and 318 repre- 
sent solid blocks, or prisms, standing upright on a level base, bd ; 
the upner surface, J a forming obiique angles with the face, b I — 

46 



362 



AMERICAN HOUSE-CARPENTER. 




Fig. 316. 



in Fig. 317 obtuse, and in Fig. 318 acute. Upon the base, de 
scribe the semi-circle, h s c ; from the centre, i, draw i 5, at right 
angles tob c ; from s, draw s x, at right angles to e c?, and from i 
draw i A, at right angles to 6 c ; make i h equal to 5 x. and join 
h and x ; then, h and .r being of the same height, the line, h x, 
joining them, is a level line. From h, draw h n, parallel to b c, 
and from 7i, draw n r, at right angles to 6 c ; join r and 5, also v 



&TAIRS. 



303 





Fig. 317. 



Fig. 318. 



and X ; then, n and x being of the same height, nx'\s^. ^evel hne ; 
and this line lying perpendicularly over r s, n x and r s must be 
of the same length. So, all lines on the top, drawn parallel to n 
X, and perpendicularly over corresponding lines drawn parallel to 
r 5 on the base, must be equal to those lines on the base ; and by 
drawing a number of these on the semi-circle at the base and 
others of the same length at the top. it is evident that a curve, j 
X I, may be traced through the ends of those on the top, which 
shall lie perpendicularly over the semi-circle at the base. 

It is upon this principle that the process at Fig. 315 and 316 
is founded. The plan of the rail at the bottom of those figures 
is supposed to lie perpendicularly under the face-mould at the top ; 
and each ordinate at the top over a corresponding one at the base. 
The ordinates, n x and r s, in those figures, correspond to n x 
and r s in Fig. 317 and 318. 

In Fig. 319, the top, e «, forms a right angle with the face, d 
c ; all that is necessary, therefore, in this figure, is to find a line 
corresponding to A :r in the last two figures, and that will lie level 
and in the upper surface ; so that all ordinates at right angles to 
d r on the base, will correspond to those that are at right angles 



364 



AMERICAN HOUSE rARPENTER. 




Fig 319 r 



to c c on the top. This elucidates Fig. 307 ; at wnich the lines, 
ft 9 and i 8. correspond to A 9 and i 8 in this figuie. 




Fig. 320. 



482. — To find the bevil for the edge of the plank. The 
plank, before the face-mould is applied, must be bevilled accord- 
ing to the angle wliich the top of the imaginary block, or prism, 
in the previous jftgures, makes with the face. This angle is de- 
termined in the following manner : draw w i, {Fig. 320,) at right 
angles to i s, and equal to i^? A at Fig. 315 ; make i s equal to t 5 in 
that figure, and join w and s ; then 5 'M? p will be the bevil required 
in order to apply the face-mould at Fig. 315. In Fig. 316, the 
middle height being below the line joining the other two, the bevil 
is therefore acute. To determine this, draw i 5, {Fig. 3.21,) a1 



STAIRS. 



3G5 




Fig. 821. 



right angles to ip, an d equal to ^ 5 in Fig. 316 ; make s i? equal 
to h win Fig. 316, and join w and i ; then w i p will be the 
bevil required in ord3r to apply the face-mould at Fig. 316. Al 
though the falling-mould in these cases is curved, yet, as the 
plank is sprung, or bevilled on its edge, the thickness necessary 
to get out the twist may be ascertained according to Art. 474 — 
taking the vertical distance across the falling-mould at the joints, 
and placing it down from the two outside heights in Fig. 315 or 
316. After bevilling the plank, the moulds are applied as at Art. 
476 — applying the pitch-board on the bevilled instead of a square 
edge, and placing the tips of the mould so that they will bear the 
same relation to the edge of the plank, as they do to the line, j Z, 
in Fig. 315 or 316. 




Fig. 322. 



483. — To ap]^ly the moulds without bevilling the plank. 
Make w p, (Fig. 322,) equal to w p aX Fig, 320, and the angle, 
bed, equal to b j I in Fig. 315 ; make p a equal to the thick- 
ness of the plank, as iv a ir Fig. 320, and from a draw a o, pa- 
rallel to w d ; from c, draw c e, at right angles to w d, and join c 



866 



AMERICAN HOUSE-CARPENTER. 



and b ; then the angle, 6 e O; on a squ are edge of the plank, hav 
ing a line on the upper face at the distance, p a, in Fig. 320, at 
which to apply the tips of the mould — will answer the same pur- 
pose as bevilling the edge. 

If the bevilled edge of the plank, which reaches from p to w^ 
is supposed to be in the plane of the paper, and the point, a, to 
be above the plane of the paper as much as a, in Fig. 320, is dis- 
tant from the line, w p ; and the plank to be revolved on p 6 as 
an axis until the line, p w, falls below the plane of the paper, and 
the line, p a, arrives in it ; then, it is evident that the point, c, 
will fall, in the line, c e, until it lies directly behind the point, e, 
and the line, b c, will lie directly behind b e. 




484. — To find the bevils for splayed work. The principle 
employed in the last figure is one that will serve to find the bevils 
for splayed work — such as hoppers, bread-trays, (fcc. — and a way 
of applying it to that purpose had better, perhaps, be introduced 
in this connection. In Fig. 323, a 6 c is the angle at which the 
work is splayed, and b d, on the upper edge of the board, is at 
right angles to a b ; make the angle, /^^', equal to a b c, and 
from/, draw /A, parallel to e a; from b, draw b o, at right an 
gles to a b ; through o, draw i e, parallel to c b, and join e and 
d ; then the angle, a e d, will be the proper bevil for the ends from 
the inside, or k d e from the outside. If a mitre -joint is le* 



STAIRS. 367 

quired, setfg, the thickness of the stuff on the level, from e to 
m, and join m and d ; then k d m will be the proper bevil for a 
mitre-joint. 

If the upper edges of the splayed work is to be bevilled, so as 
to be horizontal when the work is placed in its proper position, 
f g j^ being the same as a b c, will be the proper bevil for that 
purpose. Suppose, therefore, that a piece indicated by the lines, 
k gy gf and /A, were taken off; then a line drawn upon the 
bevilled surface from c?, at right angles to k d, would show the 
true position of the joint, because it would be in the direction of 
the board for the other side ; but a line so drawn would pass 
through the point, o, — thus proving the principle correct. So, if 
a line were drawn upon the bevilled surface from d, at an angle 
of 45 degrees to k d, it would pass through the point, n. 

485. — Another method for face-moulds. It will be seen by 
reference to Art. 481, that the principal object had in view in the 
preparatory process of finding a face-mould, is to ascertain upon it 
the direction of a horizontal line. This can be found by a method 
different from any previously proposed ; and as it requires fewer 
lines, and admits of less complication, it is probably to be preferred. 
It can be best introduced, perhaps, by the following explanation . 

In Fig. 324, _; d represents a prism standing upon a level base, 
b df its upper surface forming an acute angle with the face, 
b Z, as at Fig. 318. Extend the base line, b c, and the raking 
line, J I, to meet at/; also, extend e d and g a, to meet at k; 
from /, through k, draw / m. If we suppose the prism to stand 
upon a level floor, o f m,, and the plane, ^* g a l, to be extended 
to meet that floor, then it will be obvious that the intersection 
oetween that plane and the plane of the floor would be in the line, 
f k; and the line, /A:, being in the plane of the floor, and also in 
the inclined plane, ^'^ kf, any line made in the plane, J ^ kf 
parallel to/ A*, must be a level line. By finding the position of a 
perpendicular plane, at right angles to the raking plane, J//*: g, 
we shall greatly shorten the process for obtaining ordinates. 



S(\?> 



AMERICAN HOUSE-CARPENTER. 




Fig. 324. 



This may be done thus : fromj^ drawfo, at right angles tofm; 
extend e b to o, and g j, to t ; from o, draw o t, at right angles to 
of, and join t and/; then to/ will be a perpendicular plane, at 
right angles to the inclined plane, t g kf; because the base of 
the former, of, is at right angles to the base of the latter,/ Ar, both 
these lines being in the same plane. From 6, draw h p, at right 
angles to of or parallel tofm ; from 2?. draw p q, at right angles 
to of and from q, draw a line on the upper plane, parallel to/m, 
or at right angles to tf; then this line will obviously be drawn 
to the point, J, and the line, qj, be equal top b. Proceed, in the 
same way, from the points, s and c, to find a; and I. 

Now, to apply the principle here explained, let tlie curve, b s c^ 
{Fig. 325,) be the base of a cylindric segment, and let it be re- 
quired to find the shape of a section of this segment, cut by a 
plane passing through three given points in its curved surface: 
one perpendicularly over b, at the h'eight, bj; one perpendicu- 
larly over s, at the height, s x ; and the other over c, at the height, 
c I — these lines being drawn at right angles to the chord of the 
base, b c. Fromj, through Z, draw a line to meet the chord line 
extended to/; from 5, draw s k, parallel to b f and from a:, 
draw a; k, parallel tojf; from/ through k, draw/w; then fm 
will be the intersecting line of the plane of the section with the 



STAIRS. 



369 




Fig. 825. 



plane of the base. This Hne can be proved to be the intersection 
of these planes in another way ; from 6, through s, and from j, 
through .r, draw lines meeting at m ; then the point, m, will be 
in the intersecting line, as is shown in the figure, and also at 
Fig. 324. 

From/, draw/p, at right angles to/ m ; from b and c, and 
from as many other points as is thought necessary, draw ordinates, 
parallel to fm; make p q equal to b j, and join q and/; from 
the points at which the ordinates meet the line, qf, draw others 
at right angles to q f; make each ordinate at A equal to its cor- 
responding ordinate at C, and trace the curve, gn i, through the 
points thus found. 

Now it may be observed that A is the plane of the section, B 
the plane of the segment, corresponding to the plane, q pf,oi 
Fig. 324, and C is the plane of the base. To give these planes 
their projer position, let A be turned on ^ / as an axis until it 



STO AMERICAN HOUSE-CARPENTER. 

Stands perpendicularly over the line, qf^ and at riglit angles to 
the plane, B ; then, while A and B are fixed at right angles, let 
B be turned on the line, ]j /, as an axis until it stands perpendicu- 
larly over p/, and at right angles to the plane, C ; then the plane, 
A, will lie over the plane, C, with the several lines on one corres- 
ponding to those on the other ; the point, i, resting at Z, the point, 
n, at X, and g atj ; and the curve, g n i, lying perpendicularly 
over 6 s c — as was required. If we suppose the cylinder to be 
cut by a level plane passing through the point, Z, (as is done in 
finding a face-mould,) it will be obvious that lines corresponding 
to 9'/ and jt?/ would meet in I ; and the plane of the section, A. 
the plane of the segment, B, and the plane of the base, C, would 
all meet in that point. 

48(). — To find the face-mould for a hand-rail according to 
the principles explained in the previous article. In Fig. 326, 
a e cf is the plan of a hand-rail over a quarter of a cylinder ; and 
in Fig. 327, a b c d is the falling-mould ; / e being equal to the 
stretch-out of a df in Fig. 326. From c, draw c h, parallel to 
ef; bisect h c in i, and find a point, as b, in the arc, df (Fig. 
326,) corresponding to i in the line, h c; from i, {Fig. 327,) to 
the top of the falling-mould, draw i j, at right angles to he; at Fig. 
326, from c, through b, draw c g, and from b and c, draw bj and 
c /j, at right angles to ^ c ; make c k equal to h g at Fig. 327, 
and bj equal to i j at that figure ; from k, through j, draw k g, 
and from^, through a, drciw g p ; then ^/> will be the intersecting 
line, corresponding to fm in Fig. 324 and 325 ; through e, draw 
p 6, at right angles to g p, and from c, draw c q. parallel to gp ; 
make r q equal to h g at Fig. 327 ; ]omp and ^, and proceed as 
in the previous examples to find the face-mould, A. The joint 
of the face-mould, u v, will be more accurately determined by 
finding the projection of the centre of the plan, o, as at w ; 
joining s and w^ and drawing u v, parallel to s iv. 

It may be noticed that c k and b j are not of a length corres- 
ponding to the above directions : they are but^J the length given. 



6!i Alll#5. 



s:: 




37: 



AMERICAN ROUSF ^'ARPENTER, 




Fig.32T. 



The object of drawing these lines is to find the point, g, and that 
can be done by taking any proportional parts of the lines given, 
as well as by taking the whole lines. For instance, supposing c 
k and h j to be the fall length of the given lines, bisect one in i 
and the other in m; then a line drawn from m, through L will 
give the point, g^ as was required. The point, g^ may also bti 



STAIRS. 



373 



obtained thus : at Fig. 32Y, make h I equal to c 6 in Fig. 326 
rrom Z, draw I k^ at right angles to h c ; from j, draw J ^', parallel 
to h c ; from g, through k, draw g n ; at F\g. 326, make 6 ^ 
equal to / n in Fig. 327 ; then »- will be the point required. 

The reason why the points, a, b and c, in the plan of the rail a.i 
Fig. 326, are taken for resting points instead of e, i and/, is this : 
the top of the rail being level, it is evident that the points, a and e, 
in the section a e, are of the same height ; also that the point, i, is ol 
the same height as 6, and c as/. Now, if a is taken for a point 
in the inclined plane rising from the line g p^ e must be below 
that plane ; if 6 is taken for a point in that plane, i must be below 
it ; and if c is in the plane,/ must be below it. The rule, then, 
for taking these points, is to take in each section the one that is 
nearest to the line, g p. Sometimes the line of intersection, g p^ 
happens to come almost in the direction of the line, er : in such 
case, after finding the line, see if the points from which the 
heights were taken agree with the above rule ; if the heights 
were taken at the wrong points, take them according to the rule 
above, and then find the true line of intersection, which will not 
vary much from the one already found. 




487. — To apply the face-mould thus found to the plank. 
The face-mould, when obtained by this method, is to be applied 
to a square-edged plank, as dire^ed at Art. 476, with this differ- 
ence : instead of applying both tips of the mould to the edge of 



374: AMERICAN HOUSE-CARPENTER. 

the plank, one of them is to be set as far from the edge of the 
plank, as x^ in Fig. 326, is from the chord of the section p q — as 
is shown at Fig. 328. J., in this figure, is the mould applied on 
the upper side of the plank, jB, the edge of the plank, and C, the 
mould applied on the under side ; a h and c d being made equal 
to 5' ^ in Fig. 326, and the angle, e a c^ on the edge, equal to the 
angle, p q r^ dX Fig. 326. In order to avoid a waste of stuflf, it 
would be advisable to apply the tips of the mould, e and 6, im- 
mediately at the edge of the plank. To do this, suppose the 
moulds to be applied as shown in the figure ; then let A be re- 
volved upon e until the point, 6, arrives at g, causing the line, e 6, 
to coincide with e g : tlie mould upon the under side of the 
plank must now be revolved upon a point that is perpendicularly 
beneath e, as /; from/, draw / A, parallel to i d, and from d, 
draw d h, at right angles to i d ; then revolve the mould, C, upon 
f, until the point, A, arrives at j, causing the line,/ A, to coincide 
with fj, and the line, i d, to coincide with k I ; then the tips of 
the mould will be at k and /. 

The rule for doing this, then, will be as follows : make the an- 
gle, ifk, equal to the angle q v x, at Fig. 326 ; makefk equal 
iofi, and through A:, draw k /, parallel to ij ; then apply the 
corner of the mould, i, at k^ and the other corner d^ at the line, k I. 

The thickness of stuff* is found as at Art. 474. 

488. — To regulate the application of the falling-mould. 
Obtain, on the line, h c, [Fig. 327,) the several points, r, q,p, I 
and m, corresponding to the points, 6^, a^, z^ y, &c., at Fig. 326 ^ 
from r q p, (fee, draw the lines, r t, q u^p v, (fee, at right angles 
to he; make h .«?, r t, q u, &c., respectively equal to 6 c\ r q^ 6 
d\ &c., at Fig. 326 ; through the points thus found, trace the 
curve, s w c. Then get out the piece, g s c^ attached to the fall- 
ing-mould at several places alor.g its length, as at z^ z, ^z, (fee. 
In applying the falling-mould with this strip thus attached, the 
edge, sw c^ will coincide with the upper surface of the rail piece 



STAIR8. 



375 



before it is squared ; and thus show the proper position of tlic fall- 
iiig-mouid along its whole length. (See A?^t. 496.) 

SCROLLS FOR HAND-RAILS. 

4b9. — General rule for fDiding the size and position of tJte 
regulating square. The breadth which the scroll is to occupy, 
the number of its revolutions, and the relative size of the rcgula 
ting square to the eye of the scroll, being given, multiply the 
number of revolutions by 4, and to the product add the number 
of times a side of the square is contained in the diameter of the 
eye, and the sum will be the number of equal parts into which 
the breadth is to be divided. Make a side of the regulating 
square equal to one of these parts. To the breadth of the scroll 
add one of the parts thus found, and half the sum w4ll be the 
length of the longest ordinate. 



6 I 5 

ft 
7 

i 

I I I I 



490. — To find the proper centres in the regulating square. 
Let a 2 1 6, [Fig. 329,) be the size of a regulating square, found 
according to the previous rule, the required number of revolu- 
tions being If. Divide two adjacent sides, as a 2 and 2 1, into 
as many equal parts as there are quarters in the number of revo- 
lutions, as seven ; from those points of division, draw lines across 
the square, at right angles to the lines divided ; then, 1 being the 
first centre, 2, 3, 4, .5, 6 and 7, are the centres for the other quar 
ters, and 8 is the centre for the eye ; the heavy lines that deter- 



376 



AMERICAN HOUSE-CARPENTER 



mine these centres being each one part less in length than its pre 



ceding line. 




491. — To describe the scroll for a hand-rail over a curtail 
step. Let a b, {Fig. 330,) be the given breadth, If the given 
number of revolutions, and let the relative size of the regulating 
square to the eye be |- of the diameter of the eye. Then, by the 
rule. If multiplied by 4 gives 7, and 3, the rmmber of times a 
side of the square is contained in the eye, being added, the sum 
is 10. Divide a 6, therefore, into 10 equal parts, and set one from 
b to c ; bisect a c in e ; then a e will be the length of the longest 
ordinate, (1 d or 1 e.) From a. draw a d^ from e, draw e 1, and 
from i, draw 6/, all at right angles to a b ; make e 1 equal to e 
a, and through 1, draw 1 d, parallel to a b ; set b c from 1 to 2, 
and upon 1 2, complete the regulating square ; divide this square 
as at Fig. 329 ; then describe the arcs that compose the scroll, as 
follows: upon 1, describe d e ; upon 2, describe e f ; upon 3, 
describe/^ ; upon 4 describe g A, <fcc. ; make d I equal to the 



STAIRS. 



377 



width of the rail, and upon 1, describe Im ; upon 2, aescribe m 
Wj &c. ; describe the eye upon 8, and the scroll is completed. 

492. — To describe the scroll for a curtail step. Bisect d I. 
[Fig. 330,) in o, and make o v equal to i of the diameter of a 
baluster ; make v w equal to the projection of the nosing, and e 
a; equal to w I; upon 1, describe w y, and upon 2, describe y z • 
also upon 2, describe a; i ; upon 3, describe ij, and so around to 
z ; and the scroll for the step will be completed. 

493. — 7^0 determine the position of the balusters under the 
scroll. Bisect d I, {Fig. 330,) in o, and upon 1, with 1 o for ra- 
dius, describe the circle, oru; set the baluster at p fair with the 
face of the second riser, c", and from p, with half the tread in the 
dividers, space oiF as at o, q, r, s, t, u, &c,, as far as q-- ; upon 2, 
3, 4 and 5, describe the centre-line of the rail around to the eye 
of the scroll ; from the points of division in the circle, o r u, draw 
lines to the centre- line of the rail, tending to the centre of the 
eye, 8 ; then, the intersection of these radiating lines with the 
centre-line of the rail, will determine the position of the balusters, 
as shown in the figure. 



( 






s5 


p 


"^ 




2 






.^" 



Fig. 381. 



494. — To obtain the falling-mould for the raking part of the 
scroll. Tangical to the rail at A, {Fig. 330,) drav/ h k^ parallel to d 
a; then k a^ will be the joint between the twist and the other part 
of the scroll. Make d e^ equal to the stretch-out of d e, and upon d 

48 



378 



AMERICAN HOUSE-CARPENTPJR. 



e^^ find the position of the point, k^ as at k'^ ; at Fig. 331, make e d 
equal to e^ d in Fig. 330, and d c equal to d & in that figure : 
from c, draw c a, at right angles to e c, and equal to one rise ; 
make c h equal to one tread, and from 6, through a, draw h j , 
bisect a c in Z, and through Z, draw m q^ parallel to e h ; m q is 
the height of the level part of a scroll, which should always be 
about 3^ feet from the floor ; ease oif the angle, ni fj^ according 
to Art. 89, and draw g id n, parallel to m xj, and at a distance 
equal to the thickness of the rail ; at a convenient place for the 
joint, as i, draw i?i, at right angles to b j ; through w, draw / A, 
at right angles to e h ; make d k equal to d W in Fig. 330, and 
from A:, draw k o, at right angles to eh; at Fig. 330, make d 
h^ equal to c? A in Fig. 331, and draw h^ 6^, at right angles to d 
h^ ; then k a^ and h^ if will be the position of the joints on the 
plan, and at Fig. 331, o p and i n, their position on the falling- 
mould ; and p o i n, {Fig. 331,) will be the falling-mould re 
quired. 





/ 


/ 








/ 


tl 


e 


f < 





Fig. 832. 



495. — To describe the face-mould. At Fig. 330, from A:, draw 
k r^, at right angles to r^ d ; at Fig. 331, make h r equal to h^ r^ 
in Fig. 330, and from r, draw r s, at right angles to r h ; from 
the intersection of r 5 with the level line, m q, through i, draw s 
t ; at Fig. 330, make h^ b'^ equal to 9' z^ in Fig. 331, and join 6' 
and r"^ ; from a^, and from as many other points in the arcs, a^ I 
and k c?, as is thought necessary, draw ordinates to r^ 0?, at right 
angles to the latter ; make r b, {Fig. 332,) equal in its length and 
in its divisions to the line, r^ b^, in Fig. 330 ; from r, n, 0, />, q 



STAIRS. 



379 



and Z, draw the lines, r k, n d^ o a^ p e, ^/and I c, at right an- 
gles to r 6, and eqnal to r' k^ d^ 5^, f^ a', (fee, in i'^^^. 330 ; 
through the points thus found, trace the curves, k I and a c, and 
complete the face-mould, as shown in the figure. This mould is 
to be applied to a square-edged plank, with the edge, I b, parallel 
to the edge of the plank. The rake lines upon the edge of the 
plank are to be made to correspond to the angle, s t A, in Fig. 
331. The thickness of stuff required for this mould is shown at 
Fig. 331, between the lines s t and u v — u v being drawn pa- 
rallel to 5 ^. 

496. — All the previous examples given for finding face-moulds 
over winders, are intended for moulded rails. For round rails, 
the same process is to be followed with this difference : instead 
of working from the sides of the rail, work from a centre-line. 
After finding the projection of that line upon the upper plane, 
describe circles upon it, as at Fig. 293, and trace the sides of the 
moulds by the points so found. The thickness of stuff for the 
twists of a round rail, is the same as for the straight ; and the 
twists are to be sawed square through. 

h f V k 

Tl3^ 




380 



AMERICAN HOUSE-CARPENTER 



497. — To ascertain the form of the newel-cap from a section 
of the rail. Draw a b, {Fig. 333,) through the widest pait of 
the given section, and parallel to c d ; bisect a b in e, and through 
a, e and &, draw a i,fg, and kj, at right angles to a b ; at a con- 
venient place on the hne, f g, as o, with a radius equal to half 
the w^idth of the cap, describe the circle, i j g ; make r I equal 
to e 6 or e a ; join I and j, also / and i ; from the curve, / h, ta 
the line, I j, draw as many ordinates as is thought necessary 
parallel to f g ; from the points at which these ordinates meet 
the line, Ij, and upon the centre, o, describe arcs in continuation to 
meet op; from n, t, x, &c., draw n s, t u, &c., parallel to f g ; 
make n s, t u, &c., equal to e f to v, &c. ; make x y, &c., equal 
to z d, &CC. ; make o 2, o 3, &c., equal to o n, o t, &c. ; make 2 4 
equal to n s, and in this way find the length of the lines crossing 
am; through the points thus found, describe the section of the 
newel-cap, as shown in the figure. 




Fig 334. 



498. — To find the t^^ue position of a butt joint for the twists of 
a moulded rail over platform stairs. Obtain the shape of the 
mould according to Art. 466, and make the line a b. Fig. 334, 
equal to a c. Fig. 300 ; from t, draw b c, at right angles to a b, 
and equal in length to n m, Fig. 300 ; join a and c, and bisect a c 
in ; through o draw e f at right angles to a c, and d k, parallel 
to c b ; make o d and o k each equal to half e ^ at Fig. 300 ; 
through e and /, draw h i and ^ j, parallel to a c. At Fig. 301, 
make n a equal to e d, Fig. 334, and through a, draw r p, at right 
angles to n c ; then r p will be the true position on the face-mould 
or a butt joint, as was required. The sides must be sawn verti 



STAIRS. 



ISl 



cally as described at Ai^t. 467, but the joint is to be sawn square 
through the plank. The moulds obtained for round rails, {Art. 
464,) give the line for the joint, when ap) Jied to either side of the 
plank; but here, for moulded rails, th , line for the joint can be 
obliined from only one side. Whe j the rail is canted up, the 
joint is taken from the mould laid on the upper side of the lowei 
twist, and on the under side of the upper twist ; but when it is 
canted down, a course just the reverse of this is to be pursued. 
When the rail is r.^^. f^anted, either up or dow!-*. the vertical joint, 
obtained as at Art. 466, will be a butt joint, and therefore, in such 
a case, the process described in this article will be unnecessary 



NOTE TO ARTICLE 462. 



Platform stairs with a large cylinder. Instead of 
placing the platform-risers at the spring of the cyl- 
inder, a more easy and graceful appearance may be 
given to the rail, and the necessity of canting either 
of the twists entirely obviated, by fixing the place of 
the above risers at a certain distance within the cyl- 
inder, as shown in the annexed cul^ — the lines indi- 
cating the fa,ce of the risers cutting the cylinder at k 
and Z, instead of at p and q, the spring of the cylin- 
der. To ascertain the position of the risers, let a 6 c 
be the pitch-board of the lower flight, and c d e that 
of the upper flight, these being placed so that b c 
and c d shall form a right line. Extend a c to cut 
de in f; draw / g parallel to d b, and of indefinite 
length: draw g o at right angles to f g, and equal 
in length to the radius of the circle formed by the 
centre of the rail in passing around the cylinder; 
on o as centre describe the semicircle _;' g i ; make 
h equal to the radius of the cylinder, and describe 
on the face of the cylinder p h q ; then extend d b 
across the cylinder, cutting it in I and k — giving the 
position of the face of the risers, as required. To 
find the face-mould for the twists is simple and ob- 
vious : it being merely a quarter of an ellipse, hav- 
ing j for semi-minor axis, and the distance on the 

rake corresponding to o g, on the plan, for the semi-major axis, found thus,— extend i / u 
n« et a /, then from this point of meeting to / is the semi-major axis. 




SECTION VIL— SHADOWS. 



499. — The art of drawing consists in representing solids updii 
a plane surface : so that a curious and nice adjustment of lines is 
n ade to present the same appearance to the eye, as does the 
human figure, a tree, or a house. It is by the effects of light, in 
its reflection, shade, and shadow, that the presence of an object is 
made known to us ; so, upon paper, it is necessary, in order that 
the delineation may appear real, to represent fully all the shades 
and shadows that would be seen upon the object itself. In this 
section I propose to illustrate, by a few plain examples, the simple 
elementary principles upon which shading, in architectural sub- 
jects, is based. The necessary knowledge of drawing, prelim- 
inary to this subject, is treated of in the Introduction, from Art. 
1 to 14. 

500. — Tlie inclination of the line of shadow. This is always, 
in architectural drawing, 45 degrees, both on the elevation and the 
plan ; and the sun is supposed to be behind the spectator, and 
over his left shoulder. This can be illustrated by reference to 
Fig. 335, in which A represents a horizontal plane, and B arid C 
two vertical planes placed at right angles to each other. A rep 
resents the plan, C the elevation, and B a vertical projection 
(rom the elevation. In finding the shadow of th 3 plane, J5, the 



SHADOWS. 



383 




Fig. 885. 



line, a b, is drawn at an angle of 45 degrees with the horizon, and 
the line^ c b, at the same angle with the vertical plane, B. The 
plane, B, being a rectangle, this makes the true direction of the 
sun's rays to be in a course parallel to d b ; which direction has 
been proved to be at an angle of 35 degrees and 16 minutes with 
the horizon. It is convenient, in shading, to have a set-square 
with the two sides that contain the right angle of equal length ; 
this will make the two acute angles each 45 degrees; and will 
give the requisite bevil when worked upon the edge of the T- 
square. One reason why this angle is chosen in preference to 
another, is, that when shadows are properly made upon the draw- 
ing by it, the depth of every recess is more readily known, since 
the breadth of shadow and the depth of the recess will be equal. 

To distinguish between the terms shade and shadow, it will be 
understood that all such parts of a body as art not exposed to the 
direct action of the sun's rays, are in shade ,' while those parts 
which are deprived of light by the interpositrjn of other bodies^ 
are in shadow. 



384 



AMERICAN HOUSE-CARPENTER. 




601. — To find the line of shadow on mouldings and other ho- 
nzontally straight p-ojections. Fig. 336, 337, 338, and 339. 
lepresent various mouldings in elevation, returned at the left, m 
the usual manner of mitreing around a projection. A mere in- 
fepection of the figures is sufficient to see how the line of shadow 
IS obtained ; bearing in mind that the ray, a b, is drawn from the 
projections at an angle of 45 degrees. Where there is no return 
at the end, it is necessary to draw a section, at any place in -the 
length of the mouldings, and find the line of shadow from that. 

502, — To find the line of shadow cast by a shelf. In Fig. 340, 
A is the plan, and B is the elevation of a shelf attached to a wall. 
From a and c, draw a b and c d, according to the angle previously 
directed ; from b, erect a perpendicular intersecting c d at d ; from 
d, draw d e, parallel to the shelf ; then the lines, c d and d e, will 
define the shadow cast by the shelf. There is another method of 
finJing the shadow, without the plan, A. Extend the lower line 
o* tne shelf to/, and make c/ equal to the projection of the sh^lf 



SHADOWS. 



385 



B 




Fig. 840. 

from the wall ; from/, draw/^, at the customary angle, and from 
c, drop the vertical line, c g, intersecting f g dX g ; from g, draw 
g e, parallel to the shelf, and from c, draw c d, at the usual angle ; 
then the lines, c d and d e, will determine the extent of tlie shadow 
as before. 




Fig. 341. 



603. — To find the shadow cast hy a shelf , which is wider at 
one end than at the other. In Fig. 341, A is the p]an, and B 
the elevation. Find the point, d, as in the previous example, and 
from any other point in the front of the shelf, as a, erect the perpen- 
dicular, a e ; from a and e, draw a h and e c, at the proper angle, 
and from 6, erect the perpendicular, b c, intersecting e c in c . 

49 



386 



AMERICAN HOUSE-CARPENTER. 



from d, through c, draw d o ; then the hnes, i d and d o, will give 
the hmit of the shadow cast by the shelf. 




Fig. 342. 

504r. — To find the shadow of a shelf having one end acute or 
obtuse angled. Fig. 342 shows the plan and elevation of an 
acute-angled shelf. Find the hne, e g, as before ; from a, erect 
the perpendicular, a b ; join b and e ; then b e and e g will define 
the boundary of shadow. 




505. — To find the shadow cast by an inclined shelf. In Fig. 
343, the plan and elevation of such a shelf is shown, having also 
one end wider than the other. Proceed as directed for finding 
the shadows of Fig. 341, and find the points, d and c ; then a d 
anl ^ c will be the shadow required. If the shelf had been 



SHADOWS. 



887 



parallel in width on the plan, then the line, d c, would have been 
parallel with the shelf, a b. 







?k - 


''''""iiiiiiiiiiiiiiiiiiiiiiiii 




e c 




"lilB 



iiliii 



lilill 



Fig. 344 



Fig. 345. 



606. — To find the shadow cast hy a shelf inclined in its ver- 
tical section either upivard or downward. From a, {Fig. 344 
and 345,) draw a h, at the usual angle, and from 6, draw b c, 
parallel with the shelf; obtain the point, e, by drawing a line 
from d, at the usual angle. In Fig. 344, join e and i ; then i e 
and e c will define the shadow. In Fig. 345, from o, draw o z, 
parallel with the shelf; join i and e ; then i e and e c will be the 
shadow required. 

The projections in these several examples are bounded by 
straight lines ; but the shadows of curved lines may be found in 
the same manner, by projecting shadows from several points in 
the curved line, and tracing the curve of shadow through these 
points. Thus — 





Fig. 84T. 



388 



AMERICAN HOLSE CARPENTER. 



507. — To find the shadow of a shelf having its front edge^ a? 
end, curved on the plan. In Fig. 346 and 347, A and A show an 
example of each kind. From several points, as a, a, in the plan, 
and from the corresponding points, o, o, in the elevation, draw 
rays and perpendiculars intersecting at e, e, &c. ; through these 
points of intersection trace the curve, and it will define the shadov\'. 





^ 




2\ 


e' 


■1 

e 


Ulllliiiiiiiiiiiiiiiii 


l'll!llllllllllll:l 


iiiiiiiiiiiiiiii 




/ 




/ 



Fig. 348. 



508. — To find the shadow of a shelf curved in the elevation. 
In Fig. 348, find the points of intersection, e, e and e, as in the 
last examples, and a curve traced through them will define the 
shadow. 

The preceding examples show how to find shadows when cast 
upon a vertical plane ; shadows thrown upon curved surfaces are 
ascertained in a similar manner. Thus — 




SHVDOWS. 



389 



509. — To find the shadow cast upon a cylindrical wall by a 
vrojection of any kind. By an inspection of Fig. 349, it wiil be 
seen that the only difference between this and the last examples, 
is, that the rays in the plan die against the circle, a 6, instead of 
a straight line. 




Fig. 350. 

510. — To find the shadow cast hy a shelf upon an incli?ied 
loall. Cast the ray, a b, {Fig. 350,) from the end of the shelf to 
the face of the wall, and from b, draw b c, parallel to the shelf; 
cast the ray, d e, from the end of the shelf; then the lines, d e 
and e c, will define the shadow. 

These examples might be multiplied, but enough has been 
given to illustrate the general principle, by which shadows in all 
instances are found. Let us attend now to the application of this 
principle to such familiar objects as are likely to occur in practice. 




390 



AMERICAN HOUSE-CARPENTER. 



511. — To find the shadow of a projecting horizontal beam 
From the points, a, a, &c., {Fig. 351,) cast rays upon the wall 
the intersections, e, e, e, of those rays with the perpendiculars 
drawn from the plan, will define the shadow. If the beaid be m- 
chned, either on the plan or elevation, at any angle other than a 
right angle, the difference in the manner of proceeding can be seen 
by reference to the preceding examples of inclined shelves &c. 



iiiiiiiiii i 




IIIIIIIIIIIIIIMIIIIIII 



612. 



Fig. 352. 

To find the shadow in a recess. 



From th( point, a, 
{Fig. 352,) in the plan, and h in the elevation, draw th' rays, a c 
and h e ; from c, erect the perpendicular, c e, and from e, draw 
the horizontal line, e d ; then the lines, c e and e d, will show the 
extent of the shadow. This applies only where the back of the 
recess is parallel with the face of the wall. 




Fig. 353. 



513. — To find the shadow in a recess^ when the face of the. 
wall is inclined, and the hack of the recess is vertical. In Fig. 
353, A shows the section and B the ele ration of a recess of this 



SHADOWS. 



391 



kind. From 6, and from any other point in the hne, b a SiS a 
draw the rays, b c and a e ; from c, a, and e, draw the hcrizonta 
hnes, eg, a f, and e h ; from d and /, cast the rays, d i and/ ' 
from i, through h, draw i s ; then s i and i g will define tlic 
shadow. 

d 




Fig. 854. 



514. — To find the shadow in a fireplace. From a and h, 
(Fig. 354,) cast the rays, a c and b e, and from c, erect the per- 
pendicular, c e ; from e, draw the horizontal line, e o, and join c 
and d ; then c e, e o, and o d, will give the extent of the shadow. 



m 



imiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiifii 1 1 
n s ' 



►i 



a 



Fig. 355. 



515. — To find the shadow of a moulded window-lintel. Cast 
rays from the projections, a, o, & :., in the plan, {Fig. 355,) and 
d, e, &c., in the elevation, anc draw the usual perpendiculars in- 
tersecting the rays at z, i, and i ; these intersections connected 



392 



AMERICAN HOUSE-CARPENTER. 



and horizoiitai lines drawn fronr them, will define the shadow 
The shadow on the face of the lintel is found by casting a raf 
back frcm i to .9, and drawing the horizontal line, 5 n. 




Fig. 356. 



516. — To find the shadow cast hy the nosing of a step. Fitjrn 
fl, {Fig. 356,) and its corresponding point, c, cast the rays, a h 
and c d, and from b, erect the perpendicular, b d ; tangical to the 
curve at e, cast the ray, e f, and from e, drop the perpendicular, 
e o, meeting the mitre-line, a g, in ; cast a ray from to i, and 
from i, erect the perpendicular, i f ; from A, draw the ray, hk; 
from f to d and from d to k, trace the curve as shown in the 
figure ; from k and h, draw the horizontal lines, k n and h s ; then 
the limit of the shadow will be completed. 

517. — To find the shadow thrown by a pedestal upon steps. 
From a, {Fig. 357,) in the plan, and from c in the elevation, draw 
the rays, a b and c e ; then a will show the extent of the shadow 
on the first riser, as at ^ ; f g will determine the shadow on the 
second riser, as at 5 ; c d gives the amount of shadow on the 
first tread, as at C, and h i that on the second tread, as at D ; 
which completes the shadow of the left-hand pedestal, both on the 
plar and elevation. A mere inspection of the figure will be suf- 



SHADOWS. 



303 




Fig. 36T. 

ficient to show how the shadow of the right-hand pedestal is 
obtained. 





Fig. 358. 

518. — To find the shadow thrown on a column hy a square 
abacus. From a and h, {Fig. 358,) draw the rays, a c and h e, 
and from c, erect the perpendicular, c e ; tangical to the curve at 
d, draw the ray, d f, and from A, corresponding to / in the plan, 
draw the ray, h o ; take any point between a and/, as z, and from 
this, as also from a corresponding point, n, draw the rays, i r and 
n s ; from r, and from d, erect the perpendiculars, r s and do; 
through the points, e, s, and o, trace the curve as shown in the 
figure ; then the extent of the shadow vrill be defined. 

519. — To find the shadow thrown on a column hy a circular 
abacus. This is so near like the last example, that no explanation 
wUl be necessary farther than a reference to the preceding article 

50 



394 



AMERICAN HOUSE-CARPENTER. 




Fig. 360. 



620. — To find the shadows on the capital of a column. Thiu 
may be done according to the r)rinciples explained in the examples 
already given ; a quicker way ■>£ doing it, however, is as follows 
If we take into consideration one ray of light in connection with 
all those perpendicularly under and over it, it is evident that these 
several rays would form a vertical plane, standing at an angle of 
45 degrees with the face of the elevation. Now, we may sup 
pose the column to be sliced^ so to speak, with planes of this 



SHADOWS. 



395 




Fig. 361. 



nature — cutting it in the lines, ah, c d^ &c., {Fig. 360,) and, in 
the elevation, find, by squaring up from the plan, the lines of sec- 
tion which these planes would make thereupon. For instance : 
in finding upon the elevation the line of section, a b, the plane 
cuts the ovolo at e, and therefore / will be the corresponding point 
upon the elevation ; h corresponds with g, i with j, o with s, and 
I with h. Now, to find the shadows upon this line of section, cast 
from m, the ray, m n, from h, the ray, h o, &c. ; then that part of 
the section indicated by the letters, m f i n, and that part also be- 
tween h and o, will be under shadow. By an inspection of the 
figure, it will be seen that the same process is applied to eacli hne 
of section, and in that way the points, p, r, t, u, v, iv, oo, as alsc 
1, 2, 3, &c., are successively found, and the Imes of shadow 
traced through them. 

Fig. 361 is an example of the same capital with all the shadows 
finished in accordance with the lines obtained on Fig. 360. 

521. — To find the shadow thrown on a vertical wall by a 
column and entablature standing n advance of said wall. Oust 



390 



AMERICAN HOUSE-CARPENTEfl 




rays from a and h, {Fig. 362,) and find the point, c, as in the 
previous examples ; from c?, draw the ray, d e, and from e, the 
horizontal Hne, e f ; tangical to the curve at g and h, draw the 
rays, g j and h i, and from ^ and j, erect the perpendiculars, i I 
and j k ; from m and n, draw the rays, m f and n k, and trace the 
curve between k and /; cast a ray from o to p, a vertical line 
from p to 5, and through s, draw the horizontal liiie, sf; the 
shadow as required will then be completed. 



SHADOWS. 



397 




Fig. 363. 



Fig. 363 is an example of the same kind as the last, with all 
the shadows filled in, according to the lines obtained in the pre- 
ceding figure. 




Fig. 364. 



522 — Fig. 364 and 365 are examples of the Tuscan cornice. 
The manner of obtainii g the shadows is evident. 



398 



A.MERICAN HOUSE-CARPENTER. 




Fig. 365. 

•'523. — Reflected light. In shading, the finish and Hfe of ar; 
object depend much on reflected hght This is seen to advantage 
in Fig. 361 and on the column in Fig. 363. Reflected rays are 
lhT0^vn in a direction exactly the reverse of direct rays ; therefore. 
on that part of an object which is subject to reflected light, the 
shadows are reversed. The fillet of the ovolo in Fig. 361 is an 
example of this. On the right-hand side of the column, the face 
of the fillet is. much darker than the cove directly under it. The 
reason of this is, the face of the fillet is deprived both of direct 
and reflected light, whereas the cove is subject to the latter. 
Other instances of the effect of reflected light will le seen in the 
other examples. 



APPENDIX 



ALGEBKAICAL SIGNS. 



+^ plus^ signifies addition, and that the two quantities between which 
it stands are to be added together ; as a + ^, read a added to h. 

— , minus, signifies subtraction, or that of the two quantities between 
which it occurs, the latter is to be subtracted from the former ; as 
a — 6, read a minus b. 

X , multiplied by, or the sign of multiplication. It denotes that the two 
quantities between which it occurs are to be multiplied together ; 
as a X &, read a multiplied by 6, or a times &. This sign is usually- 
omitted between symbols or letters, and is then understood, as ab. 
This has the same meaning as a x 6. It is never omitted between 
arithmetical numbers ; as 9 x 5, read nine times five. 

— , divided by, or the sign of division, and denotes that of the two quanti- 
ties between which it occurs, the former is to be divided by the latter ; 
as a 4- &, read a divided by b. Division is also represented thus : 

~i in the form of a fraction. This sio-nifies that a is to be divided by 

^ b. When more than one symbol occurs above or below the line, 

CL 71 V 

or both, as •, it denotes that the product of the symbols above 

^ cm- ^ -^ 

the line is to be divided by the product of those below the line. 

=, is equal to, or sign of equality, and denotes that the quantity or 
quantities on its left are equal to those on its right ; as a — 6 =: r, 
read a minus b is equal to c, or equals c ; or, 9 — 5 = 4, read nine 
minus five equals four. This sign, together with the symbols on 
each side of it, when spoken of as a whole, is called an equation. 

aP- denotes a squared, or a multiplied by a, or the second power of a, 
and 

a' denotes a cubed, or a multiplied by a and again multiplied by a, or 
the third power of a. The small figure, 2, 3, or 4, &c., is termed 
the index or exponent of the power. It indicates how many times 
the symbol is to be taken. Thus, a^ = a a, a^ = a a a, a* =r a « a a. 
/ is the radical sign, and denotes that the square root of the quantity 
following it is to be extracted, and 

51 



4 APPENDIX. 

•y/ denotes tliat tbe cube root of the quantity following it is to be ex- 
tracted. Thus, -v/O = 3, and v^27 = 3. The extraction of rocti 
is also denoted by a fractional index or exponent, thus 
a^ denotes the square root of a, 
a^ denotes the cube root of a, 
a^ denotes the cube root of the square of a, &c. 



TRIGOITOMETKIOAL TEEMS. 




Fig. 86& 



In Fig. 366, where AB h the radius of the circle B C H^ draw a 
line A F^ from A^ through any point, (7, of the arc B G. From C draw 
C D perpendicular to ^ ^ ; from B draw B E perpendicular to AB ; 
and from G draw G F perpendicular to A G. 

Then, for the angle FA B, when the radius A equals unity, CD 
IS the sine ; AD the cosine ; D B the versed sine ; B E the tangent , 
G F the cotangent ; A E the secant ; and A F the cosecant. 



6 



APPENDIX. 



But if the angle be larger tlian one right angle, yet less than twc 
right angles, SiS £ A R, extend H A to K and E B io K^ and from B 
draw H J perpendicular to A J. 

Then, for the angle BAIT, when the radius A H equals unity, HJ is 
the sine ;AJ the cosine ; BJ the versed sine ; B K the tangent ; and 
A K the secant. 

When the number of degrees contained in a given angle is known, 
then the value of the sine^ cosine^ dc, corresponding to that angle, may 
be found in a table of Natural Sines, Cosines, &c. 

In the absence of such a table, and when the degrees contained in the 
given angle are unknown, the values of the sine, cosine, &c., may 
be found by computation, as follows: — Let B A Cj {Fig. 367,) be the 




Fig. 867. 

given angle. At any distance from A^ draw c perpendicular to A B. 
By any scale of equal parts obtain the length of each of the three lines 
a, 6, c. Then for the angle at A we have, by proportion, 

c 

a : c :: i-o 



1-0 



1-0 



sm. = 



COS. = 



tan. = 



1-0 



1-0 



1-0 



cot. = 



sec. = 



cosec. 



Or, in any right angled triangle, for the angle contained between the 
base and hypothenuse — 

Wher perp. divided by hyp., the quotient equals the sine. 
" base 
** perp. 
*' base 
" hyp. 
" hyp. 



(( 


" hvp., " 


(( 


(( 


" cosine. 


u 


" base, « 


(( 


« 


" tangent. 


(( 


" perp., " 


It 


(( 


" cotangent. 


il 


" base, " 


u 


u 


" secant. 


(( 


" perp., " 


u 


(( 


" cosecant. 



GLOS SARY. 



Terms not found here can be found in the lists of definitions Ln other parts of this book, 
or in common dictionaries. 



Abacus. — The uppermost member of a capital. 

Ahhatoir. — A slaughter-house. 

Abbey. — The residence of an abbot or abbess. 

Abutment. — That part of a pier from which the arch springs. 

Acanthus. — A plant called in English, beards-breech. Its leaves are 
employed for decorating the Corinthian and the Composite capitals. 

Acropolis. — The highest part of a city ; generally the citadel. 

Acroteria. — The small pedestals placed on the extremities and apex 
oi' a pediment, originally intended as a base for sculpture. 

Aisle. — Passage to and from the pews of a church. In Gothic ar- 
chitecture, the lean-to wings on the sides' of the nave. 

Alcove. — Part of a chamber separated by an esirade, or partition of 
columns. Recess with seats, &c., in g£ rdens. 

Altar. — A pedestal whereon sacrifice was offered. In modern 
churches, the area within the railing in front of the pulpit. 

Alto-relievo. — High relief; sculpture projecting from a surface so as 
to appear nearly isolated. 

AmphitLeaife -~f\ double theatre, employed by the ancients for the 
exhibition of gladiatorial fights and other shows. 

Ancones. — Trusses employed as an apparent support to a cornice 
upon the flanks o^* the architrave. 

Annulet. — A small square moulding used to separate others ; the 
fillets in the Doric capital under the ovolo, and those which separate 
the flutings of columns, are known by this term. 

AntcB. — A pilaster attached to a wall. 

Apiary. — A place for keeping beehives. 

Arabesque. — A building after the Arabian style. 

Areostyle. — An intercolumniation of from four to five diameters. 

Arcade — A series of arches. 

Arch. — An arrangement of stones or other material in a curvilinear 
form, so as to perform the office of a lintel and carry superhicumbent 
weights. 

Architrave. — That part of the entablature whi )h rests upon the 
capital of a column, and is beneath the frieze. The casing and 
mouldinirs about a door or window. 



APPENDIX. 

Archivolt. — The ceiling of a vault : the under surface of an aich. 

Area. — Superficial measurement. An open space, below the leve 
of the ground, in front of basement windows. 

Arsenal. — A public establishment for the deposition of arms and 
warlike stores. 

Astragal. — A small moulding consisting of a half-round with a fillet 
on each side. 

Attic. — A low story erected over an order of architecture. A low 
additional story immediately under the roof of a building. 

Aviary. — A place for keeping and breeding birds. 

Balcony.— An open gallery projecting from the front of a building. 

Baluster. — A small pillar or pilaster supporting a rail. 

Balustrade. — A series of balusters connected by a rail. 

Barge-course. — That part of the covering which projects over the 
gable of a building. 

Base. — The lowest part of a wall, column, &c. 

Basement-story. — That which is immediately under the principal 
story, and included within the foundation of the building. 

Basso-relievo. — Low relief; sculptured figures projecting from a 
surface one-half their thickness or less. See Alto-relievo. 

Battering. — See Talus. 

Battlement. — Indentations on the top of a wall or parapet. 

Bay-idndow. — A window projecting in two or more planes, and not 
forming the segment of a circle. 

Bazaar. — A species of mart or exchange for the sale of various ar- 
ticles of merchandise. 

Bead. — A circular moulding. 

Bed-mouldings. — Those mouldings which are between the corona 
and the frieze. 

Belfry. — That part of a steeple in which the bells are hung : an- 
ciently called campanile. 

Belvedere. — An ornamental turret or observatory commanding a 
pleasant prospect. 

Boiv-iuindow. — A window projecting in curved lines. 

Bressummer. — Abeam or iron tie supporting a wall over a gateway 
or other opening. 

Brick -nogging. — The brickwork "between studs of partitions. 

Buttress. — A projection from a wall to give additional strength. 

Cable. — A cylindrical moulding placed in flutes at the lower part of 
the column. 

Camber. — To give a convexity to the upper surface of a beam. 

Campanile. — A tower for the reception of bells, usually, in Italy^ 
separated from the church. 

Canopy. — An ornamental covering over a seat ot state. 

Cantalivers. — The ends of rafters under a projecting roof. Pieces 
of wood or stone supporting the eaves. , 

Capital. — The uppermost part of a column inclu'led betv/een the 
shaft and the architrave. 



APPENDIX. 



9 



Caravansera. — In the East, a large public building for tiie reception 
of travellers by caravans in the desert. 

Carpentry. — (From the Latin, carpentum, carved wood.) That de- 
partment of science and art which treats of the disposition, the cr 
struction and the relative strength of timber. Th^ first is called 
scriptive, the second constructive, and the last mechanical carpentry. 

Caryatides. — Figures of women used instead of columns to support 
' an entablature. 

Casino. — A small country-house. 

Castellated. — Built with battlements and turrets in imitation of an- 
cient castles. 

Castle. — A building fortified for military defence. A house with 
towers, usually encompassed with walls and moats, and having a don- 
jon, or keep, in the centre. 

Catacombs. — Subterraneous places for burying the dead. 

Cathedral. — The principal church of a province or diocese, wherein 
the throne of the archbishop or bishop is placed. 

Cavetto. — A concave moulding comprising the quadrant of a circle 

Cemetery. — An edifice or area where the dead are interred. 

Cenotaph. — A monument erected to the memory of a person buried 
in another place. 

Centring. — The temporary woodwork, or framing, whereon any 
vaulted work is constructed. 

Cesspool, — A well under a drain oi pavement to receive the waste- 
water and sediment. 

Chamfer. — Tlie bevilled edge of any thing originally right-angled. 

Chancel. — That part of a Gothic church in which the altar is placed. 

Chantry. — A little chapel in ancient churches, with an endowment 
for one or more priests to say mass for the relief of souls out of purga- 
tory. 

Chapel. — A building for religious worship, erected separately from 
a church, and served by a chaplain, 

Chaplet. — A moulding carved into beads, olives, &c. 

Cincture. — The ring, listel, or fillet, at the top and bottom of a co- 
*'lumn, which divides the shaft of the column from its capital and base. 

Circus. — A straight, long, narrow building used by the Romans for 
the exhibition of public spectacles and chariot races. At the present 
day, a building enclosing an arena for the exhibition of feats of horse- 
manship. 

Clerestory. — The upper part of the nave of a church above the 
roofs of the aisles. 

Cloister. — The square space attached to a regular monastery oi 
large church, having a peristyle or ambulatory around it, covered with 
a range of buildings. 

Coffer-dam. — A case of piling, water-tight, fixed in the bed of a 
I river, for the purpose of excluding the water while any work, such as 
a wharf, wall, or the pier of a bridge, is carried up. 

Collar-beam. — A horizontal beam fra ined between two principal 
rafters above the tie-beam. 

Lollcnade. — A range of columns. 

Columbarium. — A pigeon-house. 



10 



APPENDIX. 



Column. — A vertical, cylindrical support under the entablature o\ 
an order. 

(Jommon-rafters. — The same as jack-rafters, which see 

Conduit. — A long, narrow, walled passage underground, for secret 
communication between different apartments. A canal or pipe for the 
conveyance of water. 

Conservatory. — A building for preserving curious and rare exotic 
plants. 

Consoles. — The same as ancones, wbigh see. 

Contour. — The external lines which bound and terminate a figure. 

Convent. — A building for the reception of a society of religious per- 
sons. 

Coping. — Stones laid on the top of a wall to defend it from the 
weather. 

Corbels. — Stones or timbers fixed in a wall to sustain the timbers of 
a floor or roof. 

Cornice. — Any moulded projection which crowns or finishes the 
part to which it is affixed. 

Corona. — That part of a cornice which is between the crown- 
moulding and the bed-mouldings. 

Cornucopia. — The horn of plenty. 

Corridor. — An open gallery or communication to the different apart- 
ments of a house. 

Cove. — A concave moulding. 

Cripple-rafters. — The short rafters which are spiked to the hip-rafter 
of a roof. 

Crockets. — In Gothic architecture, the ornaments placed along the 
angles of pediments, pinnacles, &c. 

Crosettes. — The same as ancones, which see. 

Crypt. — The under or hidden part of a building- 

Culvert. — An arched channel of masonry or brickwork, built be- 
neath the bed of a canal for the purpose of conducting water under it. 
4ny arched channel for water underground. 

Cupola. — A small building on the top of a dome. 

Curtail-step. — A step with a spiral end, usually the first of the flight. 

Cusps. — The pendents of a pointed arch. 

Cyma. — An ogee. There are two kinds ; the cyma-recta, having 
the upper part concave and the lower convex, and the cyma-reversa, 
with the upper part convex and the lower concave. 

Dado. — The die, or part between the base and cornice of a pedestal. 

Dairy. — An apartment or building for the preservation o^ milk, and 
the manufacture of it into butter, cheese, &;c. 

Dead-shoar. — A piece of timber or stone stood vertically in brick- 
work, to support a superincumbent weight until the brickwork which 
is to carry it has set or become hard. 

Decastyle. — A building having ten columns in front. 

Dentils. — (From the Latin, dentes, teeth.) Small rectangular blocks 
used in the bed-mouldings of some of the orders. 

Diaslyie. — An intercolumniation of three, or, as some say, foa/ 
diameters. 



APPENDIX. 



11 



r)}p..~~Tha,t part of a pedestal included between the lase and tlw 
::jornice ; it is also called a dado. 

Dodecastyle. — A building having twelve columns in front. 

Donjon.— K massive tower within ancient castles to which the gar- 
rison might retreat in case of necessity. 

Books. — A Scotch term given to wooden bricks. 

Dormer. — A window placed on the roof of a house, the frame being 
placed vertically on the rafters. 

Dormitory. — A sleeping-room.* ' 

Dovecote. — A building for keeping tame pigeons. A columbarium. 

Echinus. — The Grecian ovolo. 

Elevation.-^-k geometrical projection drawn on a plane at right an- 
gles to the horizon. 

Entablature. — That part of an order which is supported by the co- 
lumns ; consisting of the architrave, frieze, and cornice. 

Eustyle. — An intercolumniation of two and a quarter diameters. 

Exchange. — A building in which merchants and brokers meet to 
transact business. 

Extrados. — The exterior curve of an arch. 

Facade. — The principal front of any building. 

Face-mould — The pattern for marking the plank, out of which hand- 
railing is to be cut for stairs, &;c. 

Facia, or Fascia. — A flat member like a band or broad fillet. 

Falliiig-mould. — The mould applied to the convex, vertical surface 
of the rail-piece, in order to form the back and under surface of the 
rail, and finish the squaring. 

Festoon. — An ornament representing a wreath of flowers and leaves. 

Fillet. — A narrow flat band, listel, or annulet, used for the separa- 
wion of one moulding from another, and to 'give breadth and firmness 
10 the edges of mouldings. 

Flutes. — Upright channels on the shafts of columns. 

Flyers. — Steps in a flight of stairs that are parallel to each other. 

Forum.—'ln ancient architecture, a public market ; also, a place 
where the common courts were held, and law pleadings carried on. 

Foundry. — A building in which various metals are cast into moulds 
or shapes. 

Frieze. — That part of an entablature included between the archi- 
trave and the cornice. 

Gahle. — The vertical, triangular piece of wall at the end of a rooi, 
from the level of the eaves to the summit. 

Gain. — A recess made to receive a t^non or tusk. 

Gallery. — A common passage to several rooms in an upper story. 
A long room for the reception of pictures. A platform raised on co- 
lumns, pilasters, or piers. 

Girder. — The principal beam in a floor for supporting the binding 
and other joists, whereby tlie bearing or length is lessened. 

Glyph. — A vertical, sunken channel. From their number, those io 
the Doric order are called triglyphs. 



12 APPENDIX. 

Granary. — A building for storing grain, especially that intended tfl 
be kept for a considerable time. 

Groin. — The line formed by the intersection of two arches, which 
cross each other at any angle. 

Guttce. — The small cylindrical pendent ornaments, otherwise called 
drops, used in the Doric order under the triglyphs, and also pendent 
from the mutuli of the cornice. 

Gymnasium. — Orig' aally, a space measured out and covered with 
sand for the exercise ( f athletic games*, afterwards, spacious buildings 
devoted to the mental as well as corporeal instruction of youth. 

Hall. — The first large apartment on entering a house. The public 
room of a corporate body. A manor-house. 

Ham. — A house or dwelling-place. A street or village : hence 
Notting/mm, Bucking/mm, &c. Hamlet, the diminutive of ham, is a 
small street ©r village. 

Helix. — The small volute, or twist, under the abacus in the Corin- 
thian capital. 

Hem, — The projecting spiral fillet of the Ionic capital. 

Hexastyle. — A building having six columns in front. 

Hip-rafter. — A piece of timber placed at the angle made by two ad- 
jacent inclined roofs. 

Homestall. — A man&ion-house, or seat in the country. 

Hotel, or Hostel. — A large inn or place of public entertainment. A 
large house or palace. 

Hot-house. — A glass building used in gardening. 

Hovel. — An open shed. 

Hut. — A small cottage or hovel generally constructed of earthy 
materials, as strong loamy clay, &c. 

Impost. — The capital of a pier or pilaster which supports an arch. 
Intaglio. — Sculpture in which the subject is hollowed out, so that 
the impression from it presents the appearance of a bas-relief 
Inter columniation. — The distance between two columns. 
Intrados. — The interior and lower curve of an arch. 

Jack-rafters. — Rafters that fill in between the principal rafters of a 
roof; called also com7non-r afters. 

Jail. — A place of legal confinement. 

Jamhs. — The vertical sides of an aperture. 

Joggle-piece. — A post to receive struts. 

Joists. — The timbers to which the boards of a floor or the laths of a 
ceiling are nailed. 

Keep. — The same as donjon, which see. 
Key-stone. — The highest central stone of an arch. 
Kiln. — A building for the accumulation and retention of heat, in ( i- 
der to dry or burn certain materials deposited within it. 
King-post. — The centre-post in a trussed roof 
Knee. — A convex bend in the back of a hand-rail. See Ramp. 



APPENDIX. 13 

Lactarium. — The same as dairy, which see. 

Lantern. — A cupola having windows in the Bides for lighting ao 
apartment beneath. 

Larmier. — The same as corona, which see. 

Lattice. — A reticulated window for the admission of air, rather than 
iight, as in dairies and cellars. 

Lever -hoards. — Blind-slats : a set of boards so fastened that they 
may be turned at any angle to admit more or less light, or to lap upon 
3ach other so as to exclude all air or light through apertures. 

Lintel. — A piece of timber or stone placed horizontally over a door, 
window, or other opening. 

Listel. — The same as fillet, which see. 

Lobby. — An enclosed space, or passage, communicating with the 
principal room or rooms of a house. 

Lodge. — A small house near and subordinate to the mansion. A 
cottage placed at the gate of the road leading to a mansion. 

Loop. — A small narrow window. Loophole is a term applied to the 
vertical series of doors in a warehouse, through which goods are de- 
livered by means of a crane. 

Luffer -boar ding. — The same as lever-boards, which see. 

Luthern. — The same as dormer, which see. 

Mausoleum. — A sepulchral building — so called from a very cele- 
brated one erected to the memory of Mausolus, king of Caria, by his 
wife Artemisia. 

Metopa. — The square space in the frieze between the triglyphs of 
the Doric order. 

Mezzanine. — A story of small height introduced between two of 
greater height. 

Minaret. — A slender, lofty turret having projecting balconies, com- 
mon in Mohammedan countries. 

Minster. — A church to which an ecclesiastical fraternity has been 
or is attached. 

Moat. — An excavated reservoir of water, surroundmg a house, cas 
tie or town. 

Modillion. — A projection under the corona of the richer orders, re 
sembling a bracket. 

Module. — The semi-diameter of a column, used by the architect as 
a measure by which to proportion the parts of an order. 

Monastery. — A building or buildings appropriated to the reception of 
monks. 

Monopteron. — A circular collonade supporting a dome without an 
enclosing wall. 

Mosaic. — A mode of representing objects by the inlaying of small 
cubes of glass, stone, marble, shells, &c. 

Mosque. — A Mohammedan temple, or place of worship. 

Mullions. — The upright posts or bars, which divide the lights in a 
Gothic window. 

Muniment -house. — A strong, fire-proof apartment for the keeping 
and preservation of evidences, charters, seals, &;c., called muniments 



14 APPENDIX. 

Museum. — A repository of natural, scientific and literary^ cunosities, 
or of works of art. 

Mutule. — A projecting ornament of the Doric cornice supposed tc 
represent the ends of rafters. 

Nave. — The main body of a Gothic church. 
Newel. — A post at the starting o;- landing of a flight of stairs. 
Niche. — A cavity or hollow place in a wall for tne reception of a 
tjtatue, vase, &c. 

Nogs. — Wooden bricks. 

Nosing. — The rounded and projecting edge of a step in stairs. 

Nunnery. — A building or buildings appropriated for the reception of 



Obelisk. — A lofty pillar of a rectangular form. 

Octastyle. — A building with eight columns in front. 

Odeum. — Among the Greeks, a species of theatre wherein the poets 
and musicians rehearsed their compositions previous to the public pro- 
duction of them. 

Ogee. — See Cyma. 

Orangery. — A gallery or building in a garden or parterre fronting 
the south. 

Oriel-window. — A large bay or recessed window in a hall, chapel, or 
other apartment. 

Ovolo. — A convex projecting moulding whose profile is the quad- 
rant of a circle. 

Pagoda. — A temple or place of worship in India. 

Palisade. — A fence of pales or stakes driven into the ground. 

Parapet. — A small wall of any material for protection on the sides 
of bridges, quays, or high buildings. 

Pavilion. — A turret or small building generally insulated and com- 
prised under a single roof. 

Pedestal. — A square foundation used to elevate and sustain a co- 
lumn, statue, &c. 

Pediment. — The triangular -crowning part of a portico or aperture 
which terminates vertically the sloping parts of the roof: this, in 
Gothic architecture, is called a gable. 

Penitentiary. — A prison for the confinement of criminals whose 
crimes are not of a very heinous nature. 

Piazza. — A square, open space surrounded by buildings. This 
term is often improperly used to denote a portico. 

Pier. — A rectangular pillar without any regular base or capital. 
The upright, narrow portions of walls between doors and windows are 
known by this term. 

Pilaster. — A square pillar, sometimes insulated, but more common 
ly engaged in a wall, and projecting only a part of its thickness. 

Piles. — Large timbers driven into the ground to make a secure 
foundation in marshy pla :;es, or in the bed of a river. 

Pillar. — A column of irregular form, always disengaged, and al- 



APPEND'X. 



15 



ways deviating f'rom the proportions of the orders ; wlience the distinc 
tion between a pillar and a column. 

Pinnacle. — A small spire used to ornament Gothic buildings. 

Planceer . — The same as soffit., which see. 

PhnlJi— The lower square member of the base of a column, pedes- 
tal, or wall. 

Porch. — An exterior appendage to a building, forming a covered 
appioacn to one of its principal doorways. 

Portal.— The aich. over a door or gate ; the framework of tlie gate : 
the lesser gate, when there are two of different dimensions at one en- 
trance. 

Portcullis. — A strong timber gate to old castles, made to slide up 
and down vertically. 

Portico. — A colonnade supporting a shelter over a walk, or ambu- 
latory. 

Priory. — A building similar in its constitution to a monastery or 
abbey, the head whereof was called a prior or prioress. 

Prism. — A solid bounded on the sides by parallelograms, and on the 
ends by polygonal figures in parallel planes. 

Prostyle. — A building with columns in front only. 

Purlines. — Those pieces of timber which lie under and at right an- 
gles to the rafters to prevent them from sinking. 

Pycnostyle. — An intercolumniation of one and a half diameters. 

Pyramid. — A solid body standing on a square, triangular or poly- 
gx)nal basis, and terminating in a point at the top. 

Quarry. — A place whence stones and slates are procured. 

Quay. — (Pronounced, key.) A bank formed towards the sea or on 
the side of a river for free passage, or for the purpose of unloading 
merchandise. 

Quoin. — An external angle. See Rustic quoins. 

Rahhet, or Relate. — A groove or channel in the edge of a board. 

Ramp. — A concave bend in the back of a hand-rail. 

Rampant arch. — One having abutments of different heights. 

Regula. — The band below the tsenia rn the Doric order. 

Riser. -^\u stairs, the vertical board forming the front of a step. 

Rostrum. — An elevated platform from which a speaker addresses aiv 
audience. 

Rotunda. — A circular building. 

RulUe-icall. — A wall built of unhewn stone. 

Rudenture. — The same as cahle, which see. 

Rustic quoins. — The stones placed on the external angle of a build- 
ing, projecting beyond the face of the wall, and having their edgea 
bevilled. 

Rustic-ivork. — A mode of building masonry wherein the faces of the 
stones are left rough, the sides only being wrought smooth where the 
union of the stones takes place. 



16 APPENDIX. 

Salon, or Saloon. — A lofty and spacious apartment compreher.dinc 
the height of two stories with two tiers of windows. 

Sarcophagus. — A ton:ib or coffin made of one sione. 

Scantling. — The measu.*e to which a piece of timber is to be or has 
been cut. 

Scarfing. — The joining of two pieces of timber by bolting or nail'r g 
transversely together, so that the two appear but one. 

Scotia. — The hollow moulding in the base of a column, between the 
fillets of the tori. 

Scroll. — A carved curvilinear ornament, somewhat resembling in 
profile the turnings of a ram's horn. 

Sepulchre. — A grave, tomb, or place of interment. 

Sewer. — A drain or conduit for carrying off* soil or water from any 
place. 

Shaft. — The cylindrical part between the base and the capital of a 
column. 

Shoar. — A piece of timber placed in an oblique direction to support 
a building or wall. 

Sill. — The horizontal piece of timber at the bottom of framing ; the 
timber or stone at the bottom of doors and windows. 

Sojfit — The underside of an architrave, corona, &c. The underside 
of the heads of doors, windows, &c. 

Summer. — The lintel of a door or window ; a beam tenoned into a 
girder to support the ends of joists on both sides of it. 

Systyle. — An intercolumniation of two diameters. 

Tcenia. — The fillet which separates the Doric frieze from the archi- 
trave. 

Talus. — The slope or inclination of a wall, among workmen called 
battering. 

Terrace. — An area raised before a building, above the level of the 
ground, to serve as a walk. 

Tesselated pavement. — A curious pavement of Mosaic work, com- 
posed of small square stones. 

Tetrastyle. — A building having four columns in front. 

Thatch. — A covering of straw or reeds used on the roofs of cottages, 
barns, &c. 

Theatre. — A building appropriated to the representation of drama..o 
spectacles. 

Tile. — A thin piece or plate of baked clay or other material used foi 
the external covering of a roof. 

Tomh. — A grave, or place for the interment of a human body, in- 
eluding also any commemorative monument raised over such a place. 

Torus. — A moulding of semi-circular profile used in the bases of 
columns. 
. Tower. — A lofty building of several stories, round or polygonal 

Transept. — The transverse portion "fa cruciform church. 

Transom. — The beam across a double-lighted window ; if the win 
dow have no transom, it is called a clerestory window. 



APPENDIX. 



17 



Tread. — Tha , part of a step which is included between the face of 
iti? riser and that of the riser above. 

Trellis. — A reticulated framing made of thin bars of wood for 
screens, windows, &c. 

Triglyph. — The vertical tablets in the Doric frieze, chamfered on 
.he two vertical edges, and having two channels in the middle. 

Tripod.- ~\ table or seat with three legs. 

Trochilus.— The same as scotia, which see. 

Truss. — An arrangement of timbers for increasing the resistance to 
cross-strains, consisting of a tie, two struts and a suspending-piece. 

Turret. — A small tower, often crowning the angle of a wall, &c. 

Tusk — A short projection under a tenon to increase its strength. 

Tympanum. — The naked face of a pediment, included between the 
icvel and the raking mouldings. 

Underpinning. — The wall under the ground-sills of a building. 
University. — An assemblage of colleges under the supervision of a 
senate, &c. 

Vault. — A concave arched ceiling resting upon two opposite paral- 
lel walls. 

Venetian-door. — A door having side-lights. 

Venetian-window. — A window having three separate apertures. 

Veranda. — An awning. An open portico under the extended roof 
of a building. 

Vestibule. — An apartment which serves as the medium of commu- 
nication to another room or series of rooms. 

Vestry. — An apartment in a church, or attached to it, for the pre- 
servation of the sacred vestments and utensils. 

Villa. — A country-house for the residence of an opulent person. 

Vinery. — A house for the cultivation of vines. 

Volute. — A spiral scroll, which forms the principal feature of the 
ionic and the Composite capitals. 

Voussoirs. — Arch-stones 

Wainscoting. — Wooden lining of walls, generally in panels. 

Water-table. — The stone covering to the projecting foundation oi 
other walls of a building. 

Well. — The space occupied by a flight of stairs. The space left 
beyond the ends of the steps is called the well-hocc. 

Wicket.- — A small door made in a gate. 

Winders. — In stairs, steps not parallel to each other. 

Zophorus. — The same as frieze, which see. 

Zy^tos. — Among the ancients, a portico of unusual lergth common 
I5 appropriated to gymnastic exercises. 



TABLE OF SQUARES, CUBES; AND ROOTS. 

cFrom Hutton's Mathematics.) 



No. Square. Cube. 



1 


1 


4 


8 


9 


27 


16 


64 


25 


125 


36 


216 


49 


343 


64 


512 


81 


729 


100 


1000 


121 


1331 


144 


1728 


169 


2197 


196 


2744 


225 


3375 


256 


4096 


2S9 


4913 


324 


5832 


361 


6859 


400 


8000 


441 


9261 


484 


10643 


529 


12167 


576 


13324 


625 


15625 


676 


17576 


729 


19683 


784 


21952 


841 


24389 


900 


27000 


961 


29791 


1024 


32768 


1089 


35937 


1156 


39304 


1225 


42875 


1296 


46656 


1369 


50653 


1444 


54872 


1521 


59319 


1600 


64000 


1681 


68921 


1764 


74088 


1849 


79507 


1936 


85181 


2025 


91125 


2116 


97336 


2209 


103823 


2304 


110592 


2401 


117649 


2500 


125000 


2601 


132651 


2704 


140608 


2809 


148877 


2916 


157464 


3025 


166375 


3136 


175616 


3219 


185193 


3364 


195112 


3481 


205379 


3600 


216000 


3721 


226981 


3S44 


238328 


3J69 


250047 


4!»y6 


262144 


4<J25 


274625 


4356 


287496 


1 18.) 


300763 



Sq. Root. CubeRootJi No. Square 



1-0000000 

1 4142136 

1-7320508 

2-0000000 

2-2360680 

2-4494897 

2-6457513 

2-8284271 

30000000 

31622777 

33 165243 

3-4641016 

3 61)55513 

3-74165741 

3-8729833' 

400J000()j 

4-12310561 

4-24264071 

4-358S939 

4-472l3!;o 

4-582575/1 

4-6904153 

4-7953315' 

4-898 J795[ 

S-OJOOOOOi 

5-0990195! 

5-196l5-24i 

5 2915>)26i 

5-38516481 

5-47722581 

5 56776441 

5-6568542; 

5-74458261 

5-83J9519: 

5-9160798 

6-0009000 

6-0327625' 

6-1644140 

6-2449980! 

6-3245553 

6-4031242 

6-4307407 

6-5574385; 

6-6332496' 

6-703203J; 

6-732330.) 

6-«558548; 

692320321 

70000000; 

7-0710678 

7-1414284! 

7-2111026 

7-2301099 

7-3484692' 

7-4161985 

7-4833143 

7-5493344 

7-6157731 

7-6311457 

7-7459667 

7-8102497 

7-8740079 

7-9372539 

8 0000000 

8-0622577 

8- J 240334 

8-1853523 



1000000 
1-259921 
1-44-2250 
1-537401 
1-709976 
1-817121 
1-912931 
2-000000 
2-080034 
2-154435 
2-2239801 
2-2^9429| 
2 351335: 
2-410142 
2-466212: 
2-519842i 
2-571232! 
2-6207411 
2-66S4U2 
2-714418! 
2-758J24| 
2-302039; 
2-843i67! 
2-88 W99; 
2-9240181 
2-962496! 
3-000000! 



3-036589; 

3-072317! 

3- 107232 i! 

3-1413:ii;i 

3-174302, 

3-207534 

3-23J612 

3-271066J 

3 3J1927; 

3-332222: 

3361975! 

3-391211! 

3 419952 

3 448217 

3-476027; 

3-503393 

353..3481 

35563:j3 

3-533 J48! 

3-608828 

3-634-241 

3-659306 

3634031 

3-70843J 

3-732511 

3-756286 

3-779763 

3-.S02952 

3-8258621 123 

3-843501 

3-870877 

3-892996 

3-914868 

3-936497 

3-957891 

3-979057 

4-000000 

4-020726 

4-041240 

4-061548 



68 
69 
70 
71 
72 
73 
74 
75 
76 
7 

78 
79 
80 
81 
82 
83 
84 
85 
86 
87 
88 
89 
90 
91 
92 
93 
94 
95 
96 
97 
98 
9J 
100 
101 
102 
103 
104 
105 
106 
107 
108 
109 
110 
111 
112 
113 
114 
115 
116 
117 
118 
119 
120 
121 
122 



124 
125 
1-26 
127 
128 
129 
130 
131 
132 
133 
134 



4624 

4761 

4900 

5041 

5184 

53-29 

5476 

5625 

5776 

5929 

6084 

6241 

640J 

6561 

6724 

68 ■!9 

7i 58 

7225 

7396 

7569 

7744 

7921 

8100 

8281 

8164 

8649 

8836 

9025 

9216 

9409 

9604 

9801 

10000 

10201 

10404 

10609 

10816 

11025 

11236 

11449 

11664 

11331 

12100 

12321 

12544 

12769 

12996 

13-225 

13456 

13689 

13924 

14161 

14400 

14641 

14884 

15129 

15376 

] 56-25 

15876 

16129 

16334 

16641 

16900 

17161 

17424 

17689 

17956' 



Cube 



31443: 

328509 

343000 

357911 

373248 

38901 

4U5224 

421875 

433976 

456533 

474552 

493039 

51-2000 

531441 

551388 

57178 

592704 

614125 

636058 

658503 

681472 

704969 

729000 

753571 

778883 

804357 

8305 S4 

857375 

884736 

912673 

941192 

970299 

1000000 

103J301 

1061208 

1092727 

1 12186 i 

11576-25 

1191016 

12-25043 

1-259712 

1295029 

1331000 

1367631 

1404928 

144-2897 

1481544 

1520875 

1560896 

1601613 

1643032 

1685159 

1728000 

1771561 

1815848 

1860867 

1906624 

1953125 

2000376 

2048383 

2097152 

2146689 

2197000 

2248(>9l| 

22999681 

235263^1 

2406104'' 



Sq. Root. 



CubeRoot. 



8-2462113 

8-30662391 

8-3666003' 

8-4261498; 

8-4852814 

854401)37 

8-6023253 

8-6602540 

8-7177979 

8-7749644 

8-8317609 

8-88819441 

8-9442719; 

9-0{X)()000 

9-055385l| 

9-1104336; 

9-1651514! 

9-2195445; 

9-27361851 

9-3-273791 

9-3808315 

9-4339811 

9-4363330 

9-5393929 

9-5916630 

9 64365t)8 

9-6953597 

9-7467943 

9-7979590 

9-848S578 

9-8994949 

9-9498744 

10-0000000 

10-0498753! 

10-09950491 

10-1483916 

10-1980390' 

10-24695 K8' 

10-29583-'.i; 

10-31403O4; 

10-3923>t3 

10-4403.;65' 

10-4880335 

10-53585:r-i 

10-533at)5-2' 

10-6301453; 

10-67707^:1 

10 723305;i 

10-77w32.ia! 

10-816:-)533i 

10-8627805 

10-9087 12 (, 

10-954451:: 

11-0000'OvK) 

11 -045317 10 

ll-09053v85 

11-1355287 

11-1803399 

11-22497-22 

11-2694277 

11-3137085 

ll-.35Vc;!67 

11401754;]| 

11-445.V23I 

n •4391-2531 

11-53-256261 

ll-'ST5S369l 



d 



4-081655 

4 101566! 

4-121-2851 

4-140818 

4-160168 

4-179339 

4-198336 

4 217163 

4 235824 

4-254321 

4-27-2659 

4-2iH>840 

4-308869 

4-3i6'749 

4-344481 

4-362071 

4-379519 

4-396830 

4-414005 

4 -'431048 

4-447960 

4-464745 

4 "J 8 1405 

4-49794 

4 514357 

4-530-855 

4 516336 

4-582903 

4-573S57 

4-59i701 

4010436 

4-626065 

4 641539 

4-657009 

4^72329 

4-687548 

4-702659 

4-717694 

4-732623 

4 717459 
4-7822)3 
4-776856 
4-79142 
4-805395 
4-H2J234 
4-ii345i3 
4-8433)8 
4-8P.294i 
4-878999 
4-890973 
4-9 jiS&i 
4-913635 
4-932424 
4 -946 187 
4-959676 
4-973190 
4-936631 
5-000000 
5-013293 
5-0265;:6 

5 039634 
5 052774 
5 065797 
5-078753 
5-091643 
5 104469 
5 117-230 



APPENDIX. 



19 



^0. 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


No. 


Square. 


Cul)e. 


Sq. Root. 


CubeRoot. 


135 


18225 


2460375 


116189500 


5-129923 


202 


4- ■804 


824-2403 


14-21-26704 


5-867464 


1.36 


18496 


2515156 


11-661903^ 


5-14-2563 


203 


41209 


83654-27, 14-2473063 


5-877131 


137 


18769 


2571353 


11-7046999 


5-155137 


204 


41616 


843J664 


14-2523569 


5-836765 


13S 


19044 


2623072 


11-7473401 


5-16764 1 


205 


4 202') 


8615125 


14-3173211 


5 -89636a 


139 


19321 


2635619 


11-7898261 


5-180101 


206 


42 i;s. 


8741816 


14-3527001 


5-905941 


140 


19600 


2744000 


11 -832 159;. 


5-192494 


207 


4-2819 


83(59743 


14-3374946 


5-915482 


141 


19H81 


2303221 


11-8743422 


5-204328 


203 


43264 


8998912 


1^4222051 


5-924992 


142 


20164 


2363283 


11-9163753 


5-217103 


209 


43631 


9129329 


14-4.56a323 


5-934473 


143 


20449 


2924207 


ll-958-26:)7 


5-2-29321 


210 


44100 


9-26 1000 


14-4913767 


5-943J22 


144 


2!>736 


2985934 


12 0000000 


5-241483 


211 


44521 


9393931 


14-5258390 


5-953342 


145 


21025 


3048625 


12-0415946 


5-253583 


212 


44944 


9528123 


14-5692193 


5-962732 


146 


21316 


3112136 


12-083:)460 


5-265637 


213 


45369 


9663507 


14-5945195! 5-972093| 


147 


21609 


317r)523 


12-1243557 


5-277632 


214 


45796 


9300344 


1462873J3 


5-931424 


148 


219!)4 


3241792 


12-1655251 


5-289572 


215 


46225 


9933375 


14-662S783I 5 9S0726| 


149 


22201 


33;)7949 


12-2065556 


53JI459 


216 


46656 


10077-.i96 


14-6069335 


6-000000 


150 


2250J 


3375000 


12-2474487 


5-3132J3 


217 


47089 


lu2 18313 


14-7309199 


6-009245 


151 


22 sot 


3442951 


12-2332057 


5-325074 


218 


47524 


10^330,32 


14-7618231 


6-013462 


152 
.153 


23104 


3511808 


12-3238280 


5-336803 


219 


47961 


10503459 


14-7986486 


6-027650 


23409 


3531577 


12-3693169 


5-348481 


2-20 


48400 


10643000 


14-8323970 


6-036811 


154 


23716 


36522r,4 


12-4096736 


5-360108 


221 


43S41 


10793861 


14-3660587 


6045943 


155 


24025 


3723375 


12-449899. 


5-371685 


222 


49234 


10941048 


14-89.)6644 


6-055049 


156 


2433r. 


3796416 


12-4399960 


5-333213 


223 


49729 


1103,-567 


14-9331845 


6-064127 


157 


21619 


3869393 


12-5299641 


5-394691 


224 


50176 


1 1239424 


14-963f5295 


6-07317S 


153 


24964 


3944312 


12-5698051 


5-406120 


225 


50625 


11390625 


15-OjOOOjO 


6-0(32202 


159 


25281 


4019679 


12-6995202 


5-417501 


226 


51076 


11513176 


15 0332961 


6-091190 


160 


2:>600 


4096000 


12-6491106 


5-428335 


227 


51529 


1 1697033 


15 0665192 


6-109170 


16! 


2;)921 


4173281 


12-6385775 


5-440122 


223 


51984 


11852352 


15 •0996639 


6109115 


16-i 


i.'.2i4 


4-^5152 S 


12-7279221 


5-451352 


229 


52441 


12008939 


15- 1327460 


6-113J33 


16:] 


265:59 


433J747 


12-7671453 


5-462556 


230 


52900 


12167000 


15-1657509 


6126923 


mi 


>;68Ji; 


4410944 


12-8062485 


5-473704 


231 


5.i361 


12323391 


15 1936342 


6-1357J2 


165 


y-7225 


4492125 


12-84523-26 


5-434897 


! 232 


538-24 


12487168 


15-2315462 


6-144634 


166 


?;7556 


4574296 


12-8840937 


5-495365 


1 233 


54-289 


1-2649337 


15-2643375 


6-153440 


167 


27839 


4657463 


12 9228480 


5-506878 


234 


5475:5 


12812904 


15 -2370585 


6-162240 


168 


23224 


4741632 


12-9614314 


5 517843 


1235 


55-2-25 


12^77375 


15-3297097 


6-171006 


169 


28561 


4826809 


13 0000000 


5 •523775 


i 236 


55696 


13144256 


15 3322915 


6-179747 


170 


28yoo 


4913000 


130331048 


5-539653 


237 


56169 


1331-2053 


15-3943043 


6-183463 


171 


29211 


S)00211 


13-0766968 


5-550499 


! 233 


55644 


13J81272 


15-4272486 


6197154 


172 


29>84 


5083448 


13-1143770 


5-561293 


239 


57121 


l3of)VJl'j 


15-4596248 


6-20582;^ 


1 173 


29929 


5177717 


13-1529464 


5-572055 


240 


57600 


1332 4000 


15-4919334 


6214465 


174 


30276 


5268024 


13-190.)060 


5 532770 


241 


53031 


139.'7521 


15-5241747 


6223084 


175 


30625 


5359375 


13-2-287566 


5-593445 


i 242 


53564 


14172488 


15-5563492 


6-2316 iO 


176 


309% 


5451776 


13-2664992 


5-604079 


1 243 


59J4y 


14348907 


15-5334573 


6-240251 


177 


31329 


5545233 


13-3041347 


5-614672 


1 244 


59536 


14526784 


15-6204994 


6-243309 


178 


31i;84 


5539752 


13-3416641 


5-625226 


1 245 


60025 


14706125 


15-6524753 


6-257325 


179 


32041 


5735339 


13-379;)SS2 


5-635741 


! 246 


60516 


14336936 


15-6343371 


6-265327 


180 


32400 


5832000 


13-4 Iti 4079 


5-646216 


1 247 


6100J 


150,50223 


15-7162336 


6-274305 


181 


32761 


5929741 


13-4536-240 


5-656653 


1 248 


61504 


1525:992 


15-7489157 


6-232761 


182 


33121 


6023568 


13 4907376 


5-667051 


i 249 


620 J 1 


15433240 


15-7797338 


6-291195 


183 


33t'39 


6128487 


13-5277493 


5-677411 


1 250 


62500 


15,525000 


15-8113383 


6-299605 


181 


33J.6 


6229504 


13-5646600 


5-637734 


j 251 


63001 


15313^51 


15-8429795 


6-3J7994 


185 
18r> 
187 


34225 


6331625 


136014705 


5-693019 


252 


63504 


16003008 


15-8745079 


6-316360 


34596 


6434356 


13-6331317 


5-7032(57 


253 


64009 


16194277 


15-9!59737 


6-3247J4 


34969 


6539203 


13-6747943 


5-718479 


1 254 


64516 


16387064 


15-9373775 


6-333026 


188 


35344 


6644672 


13-7113092 


5-723654 


255 


65025 


16.531375 


15 9687194 


6-341323 


189 


35721 


6751269 


13-7477271 


5-733794 


256 


65536 


16777216 


16 00J,)090 


6-349604 


190 


36100 


6859000 


13-7840433 


5-748397 


257 


63049 


16,)74593 


16 .<3 12 195 


6 357861 


191 


364.-(l 


6967871 


13-82L)275a 


5-758965 


253 


66564 


17173512 


13-0623734 


6-366097 


192 


36864 


7077838 


13-8564065 


5-763998 


1 259 


67031 


17373979 


16-0934769 


6-374311 


1J3 


372i9 


7189057 


13-8924440 


5-778996 


j 260 


67600 


17576JOO 


16-12t5155 


6-382594 


194 


37.'.36 


7301334 


13-9283883 


5-783'J60 


261 


6<121 


17779.531 


16 1554944 


6-390676 


195 


33025 


7414875 


13-964-2400 


5-798390 


262 


6^614 


17981723 


13-1864141 


6-393323 


196 


33416 


7529536 


14-0000000 


5-808736 


263 


691(59 


18191447 


16-2172747 


6-406953 


197 


3 Yd,)3 


7615373 


14-035 663 S 


5-818643 


261 


696915 


13399744 


1524S0763 


6-415J(i9 


19-8 


39204 


7762392 


14-0712473 


5-823477 


265 


70225 


186J9625 


16-2738206 


6-42315^ 


19:) 


3 J.. 01 


7880599 


14 1067360 


5-833272 


26:5 


70756 


18321096 


16-3095964 


6-431223 


200 


40000 


8000000 


14-1421356 


5-8 43935 


267 


71239 


19034163 


16-3401 M6 


6-139277 


201 


40101 


8120601 


14-1774469 


5-857766 


: 263 


71824 


19248832 


16-3707055 


6-447306 



53 



20 



APPENDIX 



Na 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


No. 


Square. 


Cube. 


Sq Root. 


CubeR'^ot- 


269 


72331 


19465109 


16-4012195 


6-455315 


336 


112896 


37933056 


18-3303028 


6-952053 


270 


72900 


19633000 


16-4316767 


5-463304' 


337 


113559 


38272753 


18 3575598 


6-9539 i 3 


zil 


/3141 


19902511 


16-46-20776 


6-471274 


333 


114244 


38514472 


18-3347763 


6-965320 


272 


73984 


20123648 


16-4924225 


6-479224 


339 


114921 


38908219 


18-4119526 


6-972683 


273 


74529 


20346417 


1 6-52271 16 


6-487154 


340 


115)00 


393"4000 


18-4390839 


6-979532 


274 


75076 


20570824 


16-5529454 


6-495065 


341 


116231 


39651821 


184661853 


6-986363 


275 


75625 


20796875 


15-5331240 


6-502957 


342 


116954 


40001688 


18-4932420 


6-993191 


276 


76176 


21024576 


16-6132477 


6-510830 


343 


117649 


40353607 


18-5202592 


7 000000 


277 


76729 


21253933 


16-6433170 


6:5l35i4 


344 


1 18336 


40707584 


18-5472370 


7-005796 


278 


77234 


21484952 


166733320 


6 5-25519 


345 


119025 


41053625 


18-5741756 


7-013579 


279 


77311 


21717639 


16-7032931 


6-531335 


346 


119716 


41421736 


186010752 


7-020349 


2S0 


78400 


21952000 


16-7332005 


6-542133 


347 


120409 


41781923 


18-6279350 


7027106 


281 


78951 


22188041 


16-7630546 


6-549912 


343 


121104 


42144192 


18 6547581 


7-033350 


282 


79524 


2242576S 


16-7923555 


6-557672 


349 


121801 


42508549 


18-5315417 


7-040531 


233 


80089 


22665187 


16-8226033 


6-5:i54l4 


350 


1-2-2500 


42875 JOO 


18-7082369 


7-047299 


234 


80656 


229063M 168522995 


6-573139 


351 


123-201 


43243551 


18-7349940 


7-054004 


2i5 


81225 


23149125 


16-8319430 


6-530344 


352 


123904 


43614208 


18751663) 


7-060697 


286 


81796 


23393656 


16 9115345 


6 538532 


353 


124609 


4398 )977 


18-7832942 


706737/ 


287 


82369 


23639903 


16-9410743 


6-596202 


354 


125316 


44361854 


188148877 


7074044 


283 


82944 


23387872 


16-9705627 


6-503354 


355 


126025 


44738875 


18-84M4.37 


7-080699 


239 


83-121 


24137569 


170000000 


6-611489 


356 


126736 


45118016 


18-8679523 


7-087341 


290 


84100 


24389000 


17-0293864 


6-619106 


357 


127449 


45499293 


18-8944436 


7093971 


291 


84681 


24642171 


17-0537221 


6-526705! 


353 


128164 


45882712 


18-9208379 


7-100533 


292 


85264 


24897083 


17-0880075 


6-63i2S7 


359 


128881 


46263279 


18-947-2953 


7107194 


293 


85349 


251^3757 


17-1172428 


6641852 


350 


129600 


46653000 


18-9736660 


7-113787 


294 


86136 


25412184 


17-1464232 


6-549400 


361 


130321 


47045381 


19-0000000 


7-120367 


295 


87025 


25672375 


17-1755640 


6-656930 


352 


131044 


47437928 


19-0262976 


7126936 


296 


87616 


25934336 


17-20465a5 


6-664444 


353 


131759 


47832147 


19-0525589 


7-133492 


297 


8^209 


25198073 


17-2335379 


6-671940 


354 


132495 


48228544 


19-0737840 


7-140037 


293 


88804 


264G3592 


17-2626765 


6-679420 


355 


133225 


43627125 19-1049732 


7-146569 


299 


89101 


26730899 


17-2916165 


6-686883 


356 


133956 


49027895 


191311265 


7 15309) 


300 


90000 


2700000-0 


17-3205081 


6-694329 


367 


L34639 


49430853 


19-1572441 


7-159599 


301 


90601 


27270901 


17-3493516 


6-701759 


368 


135424 


49835032 


lH-1833251 


7165096 


302 


91204 


27543608 


17-3781472 


6-709173 


359 


136161 


50243409 


19 2093727 


7- 17-2531 


303 


91809 


27818127 


17-4068952 


6-716570 


370 


136900 


50553000 


19-2353341 


7179054 


304 


92416 


28094464 


17-4355958 


6-723951 


371 


137541 


51064811 


19-2513513 


7185516 


305 


93025 


28372625 


17-464-2492 


6'731316 


372 


138384 


51478848 


19-2373)15 


7-191965 


336 


93636 


23652616 


17-49285571 6-733654 


373 


1391-29 


51895117 19-3132079 


7-198405 


307 


94249 


28934443 


17-5214155 


6-745997 


374 


139876 


52313524 19-3390796 


7-204832 


308 


94864 


29218112 


17-5499288 


6-753313 


375 


140525 


52734375 19-3549167 


7-211248 


309 


95481 


29503529 


17 5783958 


6-760614 


376 


141376 


53157376 19-3307194 


7-217652 


310 


96100 


29791000 


17-6058169 


6-767399 


37- 


142129 


53532633 19-4164878 


7-224045 


311 


96721 


30080231 


17-6351921 


5-775169 


3^8 


142884 


540101521 19-4422221 


7-230427 


312 


97344 


30371328 


17-6635217 


6-782423 


379 


143541 


544399391 19-4679223 


7-236797 


313 


97969 


30664297 


17-6918050 


6-789661 


380 


144400 


54872000 19-4935337 


7-243156 


311 


98596 


30959144 


17-72004511 6-796834 


331 


145151 


55306341 195192213 


7-249504 


315 


99225 


31255875 


17-7482393 


6-804092 


332 


145924 


55742963 19-5448203 


7-255341 


31(i 


99856 


31554496 


17-7753338 


5 811235 


353 


146639 


56181887 19-5703353 


7-252167 


317 


100489 


31855013 


17-8044933 


6-818462 


1 334 


147456 


55623104 19-5959179 


7-263482 


318 


101124 


32157432 


17-8325545 


6-8-25624 


i 335 


148225 


57066625 19-6214169 


7-274786 


319 


101761 


32461759 


17-8605711 


5-832771 


336 


148996 


57512455! 19-64683-27 


7-281079 


320 


102400 


32763000 


17-8835438 


6-839904 


337 


149769 


57960603 19 6723155 


7-287362 


321 


103041 


33076161 


17-91547-29 


6-847021 


3S8 


150544 


58411072' 19-6977155 


7-293633 


322 


103584 


33336248 


17-9443584 


5-854124 


! 339 


151321 


5S353869 19-7-230829 


7-299894 


323 


104329 


33598267 


17-9722008 


6-861212 


, 390 


152100 


59319000 


19-7434177 


7-306144 


324 


104976 


34012224 


18-0000000 


6-868235 


i 391 


152831 


59776471 


19-7737199 


7 312333 


325 


105625 


34323125 


18-0277564 


6-875344 


! 392 


153654 


60236233 


19-7989899 


7-318611 


326 


106276 


34645976 


18-0554701 


6-882389 


i 393 


154449 


60693457 


19-8242276 


7-324829 


327 


106929 


31965783 


180331413 


6-889419 


394 


155236 


61162984 


19-8494332 


7 3310.37 


328 


107584 


35287552 


18-1107703 


6-8964X5 


i 395 


li6025 


61629875 


19 •8746069 


7-337234 


329 


108241 


35611239 


181353571 


6-903436 


; 396 


15-816 


62099135 


19-8997487 


7-343420 


33t 


108900 


35937000 


181659021 


6-910423 


i 397 


157609 


62570773 


19-9-243538 


7-349597 


331 


109551 


35264691 


18-1934054 


6-917396 


3;)8 


158404 


63044792 


19-9499373 


7-355752 


332 


110224 


36594368 


18-2208672 


6-9-24356 


I 399 


159201 


63521199 


19-9749844 


7361918 


333 


110839 


36926037 


1 18-2482576 


6-9313)1 


' 400 


150000 


64000000' 20-0000000 


7-363063 


331 


111556 


37259704 


1 18-2756659 


6-93S232 


401 


150801 


644812011 20 0249844 


7-374198 


335 


112225 


37595375 


! 18-3030052 


6-945150 


402 


161604 


64964308 


1 20 ^499377 


7-330323 



APPENDIX. 



21 



•Vc. 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


No. 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


4C3 


lt.2409 


65450327 


20-0748599 


7335437 


470 


22j900 


1033230JO 


21-6794334 


7-774930 


404 


1632 IB 


P5939264 


200997512 


7-392^42 


471 


221841 


104437111 


21-70^534. 


7-73049.^ 


405 


164025 


C643J125 


201246118 


7-398636 


: 472 


222734 


105154048 


21-7-255510 


7-7859.3 


406 


164836 


66923416 


20-1494417 


7-404721 


473 


223729 


105323817 


21-7485532 


7-791487 


4071 165 49 


67419143 


20-1742410 


7-410795 


474 


224676 


106496424 


21-7715411 


7-795974 


408i I .6464 


67917312 


20-1990099 


7-416859 


475 


225525 


107171875 


21 -7944 J 47 


7-302454 


409 


167281 


68417929 


2J-2237484 


7-422914 


476 


225576 


107850176 


21-8174242 


7-8079-25 


410 


168100 


68921000 


20-2434567 


7-423959 


477 


227529 


108531333 


21 -84^3297 


7-313389 


411 


1B8921 


G9426f31 


20-2731.349 


7-434994 


478 


223484 


109215352 


21-8632111 


7-813846 


412 


16J744 


699341 23 


20-2977331 


7-441019 


479 


229441 


109902239 


21-8850535 


7-824-294 


413 


170559 


70444^97 


203224014 


7-447034 


430 


2304 JO 


1105920UO 


21-908j023 


7-829735 


4H 


171395 


70957944 


20-3469899 


7-453040 


431 


231361 


111284641 


21-93171^2 


7-335169 


415 


172225 


71173375 


20-3715488 


7-459035 


432 


232324 


111930163 


21-9544934 


7-840595 


41 


17305'. 


71991296 


20-3960731 


7-465022 


433 


233289 


112678537 


21-97 7z6ii^ 


7846013 


417 


173H89 


72511713 


20-4205779 


7-470999 


434 


234256 


1133/9^04 


22-0000000 


7-851424 


418 


174724 


73034632 


20-4450483 


7-476966 


435 


235225 


114084125 
114791256 


•22-02^7155 


7-35582 , 


419 


175561 


73550059 


20-4694395 


7-4829-24 


486 


236196 


22 0454077 


7-862224 


420 


176400 


7403S0O0 


20-4939015 


7-483872 


437 


23/169 


115501303 


■i5-2-068J7t;5 


7-867513 


421 


177241 


74618461 


20-5182845 


7-494811 


438 


238144 


116214272 


220jO/22o 


7-;i72J94 


422 


178084 


75151448 


20-5126336 


7-5J0741 


4S9 


^39121 


116930169 


22 li334U 


7 ■;-! 7836,3 


423 


178929 


75636967 


20-5669638 


7-506661 


490 


240100 


117649000 


22 1359i3 . 


7-333735 


424 


179776 


76225024 


20-5912603 


7-512571 


491 


241031 


11837u771 


2-2-l5351jj 


7-8390J5 


425 


180625 


76765625 


20-6155281 


7-513473 


492 


212064 


1190.^5488 


22- 131073 J 


7-394447 


426 


181476 


77303776 


20-6397674 


7-524365 


493 


243049 


Il9823i57 


22 2o35o33 


7-899792 


427 


182329 


77854483 


20-663J783 


7-530243 


494 


244036 


120553784 


22-2261 10 i 


7-905129 


428 


183184 


73402752 


20-0881609 


7-535122 


495 


245025 


121287375 


22-2485955 


7-910460 


429 


1840 »1 


78953539 


20-7123152 


7-541987 


496 


246016 


122023J36 


222710575 


7-915783 


430 


184900 


79507000 


207364414 


7-547842 


497 


247009 


122753473 


22'293496j 


7-921099 


431 


18^761 


80062991 


20-7605395 


7-553639 


498 


248004 


123505992 


22-3l5913j 


7-925403 


432 


1866*24 


80621568 


20-7846097 


7-55J526 


499 


249001 


124-251499 


22-33^3079 


7-93i7lb 


433 


187489 


81182737 


20-8086520 


7555355 


500 


250000 


125000000 


22-3605798 


7-9370.'5 


434 


183356 


81746504 


20-8326667 


7-571174 


501 


251001 


125751501 


22-3330293 


7-942293 


435 


189225 


82312875 


20-8566535 


7-576985 


502 


252004 


126506008 


22 4053555 


7-947574 


436 


190096 


82881856 


20-8806130 


7-582786 


503 


253009 


127263527 


22-4276615 


7-952848 


437 


190969 


83453453 


20-9045450 


7-583579 


504 


254016 


128024064 


22-4499443 


7-953114 


43S 


191844 


84027672 


20-9284495 


7-5J4363 


505 


255025 


12378762) 


22-4722051 


7-963374 


439 


192721 


84604519 


20-9523268 


7600133 


506 


256036 


129554216 


22-4944433 


7-968627 


440 


193600 


85134000 


2J-976i770 


7-605905 


507 


257049 


1303^:^343 


22-51b6505 


7-973373 


441 


194481 


8576-)121 


21-0000000 


7-611663 


508 


253064 


131090512 


22 5338553 


7-9791U 


442! 195364 


86350388 


21-0237960 


7-617412 


509 


259081 


131872229 


22 5u 10283 


7-9343 i^ 


443 196249 


86938307 


21-0475652 


7-623152 


510 


260100 


I3255l0a0 


22-533 179o 


7-939570 


444 


197136 


87528334 


210713075 


7-623884 


511 


261121 


133432331 


22-6053091 


7-994788 


445 


198025 


88121125 


21-0950231 


7-634607 


1 512 


262144 


134217728 


22-6274170 


8 000001. 


446 


198916 

199809 


88716536 


21-1187121 


7640321 


513 


263169 


135005697 


22-6 495 J33 


8-0052; !0 


447 


89314623 


211423745 


7-646027 


514 


264196 


135796744 


22-6715581 


8-01 '40:^ 


448 


200704 


89915392 


21-1660105 


7-651725 


515 


265225 


13659U375 


22-5936114 


8-01559.> 


449j 201601 


90518349 


21-1896201 


7-657414 


516 


266256 


137383096 


22 7156334 


8-020779 


450 202500 


91125000 


21-2132034 


7 663094| 517 


267289 


133 18^4 13 


2^-7376340 


8-02595/ 


45 i| 203401 


91733851 


21-2357606 


7-663766; 518 


263324 


138991832 


22-75 jol3i 


8-031 lii9 


4521 204304 


92345403 


21-2502916 


7-674430 519 
7-680086 520 


269361 


139798359 


r2-78l57l5 


303u2j. 


453: 2U5209 


92959677 


21-2837967 


270400 


140608000 


22-8u35085 


8-041451 


454! 206116 


93576664 


21-3072758 


7-685733! ! 521 


271441 


141420761 


22-8^54^44 


8-046bw3 


45 j; 207025 


94196375 


21-3307290 


7-69l3/2ij 522 


272434 


142236648 


2284731;,3 


8-051748 


4561 207936 


94818816 


21-3541565 


7-6970021 523 
7-702625 524 


273529 


143055667 


22-8691933 


8 056336 


4571 208S49 


95443993 


213775533 


•^74576 


143377824 


22-8910463 


8 06 2018 


458; 209764 


96071912 


21-4009346 


7-70823911 5-25 


275625 


144703125 


22-^128785 


8-067143 


4591 210681 


96702579 


21-4242353 


7 -71 3345 |i 526 


276676 


145531576 


22-y3i6 .9;^ 


8-072.62 


460' 211G00 


97335U00 


21-4476106 


7-719443' 527 


277729 


145363183 


:^-2-95,>43 .6 


3-U/7374 


46 1| 212521 


97J72181 


214709106 


7-725032:; 528 


273734 


14719795^ 


22-y73;i500 


8 032430 


462 213444 


98 il 1128 


21-4941853 


77306141! 529 


279341 


148035389 


23 0000000 


8 u3757y 


463 2 1436 J 


99252347 


21-5174318 


7-736183'; 5.30 


5^30900 


143877000 


230^17^8.> 


8-09267, 


464i 215296 


99897344 


21-5406592 


7-741753; 531 


281961 


149721291 


23-0434372 


8-097759 


465 216225 


100544625 


21-5633537 


7-747311 'Siz 


233024 


150558768 


-23-065 1-25 i 


8- 102 S3., 


466 217156 


101194696 


21-5370331 


7-752861 : 533 


234089 


1514194;>7 


23 08o7y^8 


8-107913 


46?; 218.>89 
468 219024 


101847563 


21-6101828 


7-753402;' 534 


^85156 


152273304 


■^3 lu84400 


8-1129^., 


10i503232 


21-6333U77 


7-763.361 535 


286225 


153130375 


23- 13. 'Ob 70 


8-l)>0H 


469 ! 219961 


10 U6 1709 


21-6564078 


7-769462i 536 


287296 


153990056 


23-1516-38 


8-l23d9 



22 



APPENDIX. 



No. 
537 


Square. 


Cube. 


Sq. Root. jCubeRoot- 


^No. 


Square. 


Cube. 


Sq. Ret. 


CuteKuoi. 


288369 


154854153 23 1732605 


8-12:ll45;i 5,. 


3 4316 


220348864 


24-5764115 


8 453;>28 


£33 


239444 


1557208721 23- 1948270 


8-133137ii 605 


3 .6020 


221445125 


24-5967478 


8-457691 


539 


29,i521 


156590819' 23-2163735 


8-1332-231 606 


d'UZct 


222545016 


24-6170673 


8-462348 


540 291600 


157464O0O; 23 ^379001 


8143253:1 607 


36844 t 


223643543 


24-6373700 


8-467;-0(^ 


541 292681 


158340421 1 23-2594067 


8-148276! 608 


369664 


22475.^712 


24-6576560 


8-471647 


542 


293764 


i 59220088 23 2^)3935 


8-153294 609 


3*0831 


225866529 


24-6779254 


8-47628-9 


543 


294849 


i6OlO30O7| 23-3023604 


8-153305 610 


372100 


r26981000 


24 6981731 


8-480926 


544 


295936 


160989184' 23-3-233076 


8-163310 611 


373321 


22309 J 131 


24-7184142 


8-485558 


545 


297025 


161873625 


23-3452351 


8-158309! 612 


37455^ 


2^9220928 


24-7385338 


8490185 


546 


298116 


162771336 


23-3665429 


8-173302 


613 


3757"9 


230346397 


24-7538358 


8-494806 


547 


299209 


163667323 


23-3830311 


8-178239 


614 


376996 


231475544 


24-7790234 


8-499423 


548 


300304 


164566592 


23-4093998 


8-183269 


615 


37.5225 


232603375 


24-7991935 


8-504035 


549 


301401 


165469149 


23-4307490 


8-188244 


i 616 


379 i56 


233744396 


24 -8 193 4-73 


8-508642 


550 


302500 


166375000 


23-4520788 


8-193213 


617 


330689 


234885113 


24-8394347 


8-513243 


551 


3J3601 


167284151 


23-4733392 


8-198175 


! 618 


381924 


236029032 


24-8596058 


8-517840 


552 


3.)4704 


163196608 


23-4946802 


8-203132 


619 


333161 


237176659 


24-8797106 


8-522432 


553 


3J5809 


169112377 


23-5159520 


8-203032 


620 


33 MOO 


2333281 iOO 


24-8997992 


8-5-27019 


554 


306916 


170031464 


23-5372045 


8-213027 


621 


335641 


239433061 


24-9198716 


8-531601 


555 


308025 


170953875 


23 5534380 


8-217966 


622 


38683; 


240641848 


24-9399278 


8-535178 


556 


309136 


171879616 


23-5-96522 


8-2-22893 


623 


■68 -U\, 


241304367 


24-959'. 67;> 


8-540750 


557 


310249 


172808693 


23-6003474 


8-227825 


1 624 


339376 


242970624 


24-9799920 


8-545317 


55S 


311364 


173741112 


23 6220236 


8-^32/46 


625 


3J0625 


214140625 


25-0000000 


8-549.880 


559 


312481 


174676879 


23-6431808 


8-237661 


626 


391876 


245314376 


25-019992;) 


8-554437 


560 


313600 


175616000 


23-6543191 


8-242571 


i 627 


393129 


246491833 


25-0399681 


8-553990 


5r, 1 


314721 


176558481 


23-6854386 


8-217474 


1 623 


394334 


247673152 


25-0599282 


8-5,53538 


562 


315344 


177534328 


23-7065392 


8-252371 


! 629 


395641 


248858189 


25-0793724 


8-5(38081 


563 


316969 


178453547 


23-7276210 


8-257263 


630 


396900 


250047000 


250993003 


8-572619 


561 


318096 


17941)6144 


23-7486842 


8-^62149 


631 


3J3161 


251239591 


25-1197134 


8577152 


565 


319225 


180362125 


23-7697285 


8-267029 


! 632 


3J9424 


252435958 


25 1396102 


8-581681 


566 


320356 


181321496 


23-7907545 


8-271904 


! 633 


40068:* 


253636137 


25 1594913 


8-536205 


567 


321489 


182284263 


23-8117618 


8-276773 


634 


401956 


254810104 


25-1793566 


8'-500724 


568 


322624 


183251)432 


23-83275.6 

23-M537209 


8-281635 


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403225 


256047375 


25-199-2063 


8-595-233 


569 


323761 


184220009 


8-286493 


636 


404*96 


257259456 


25-2190404 


8-599748 


570 


324900 


185193000 


23-8746728 


8-2J1344 


1 63? 


405769 


258474353 


25 2333539 


8-604252 


571 


326041 


186169411 


23 8J56063 


8-2,;6190 


638 


407044 


259694072 


25-2585619 


8-608753 


572 


327184 


187149248 


23-9165215 


8-3)l03J 


639 


403321 


260917119 


25 2734493 


8-613-248 


573 


328329 


188132517 


23-9374184 


8-3J5865 


640 


409600 


262444000 


25.2982213 


8-617739 


574 


329476 


189119224 


23-9532 J71 


8-310694 


641 


410881 


2(33374721 


25-3179778 


8-6-22225 


575 


330625 


190109375 


23-9791576 


8-31551? 


642 


412164 


264609283 


25-3377189 


8-626706 


•676 


331776 


191102976 


24-0000000 


8-32L)3d' 643 


413449 


265347707 


25-3574447 


8-631183 


577 


332929 


192100033 


24-0208243 


8-325147li 644 


414736 


267089984 


25-3771551 


8-635655 


578 


334034 


193100552 


24-0416306 


8-32J'.)51j! 645 


416025 


2583,36125 


25-3963502 


8.640123 


579 


335241 


194104539 


24-0624188 


8-334755i 646 


417316 


269586136 


25-4165301 


8-644585 


580 


335400 


195112000 


24-0331891 


8-339551' 647 


418509 


270340023 


25-4351947 


8-649044 


581 


337561 


196122941 


24-1039416 


8-344341 648 


419904 


272097792 


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8-653497 


582 


333724 


197137368 


24-1246762 


8-349126!; 649 


421201 


273359449 


25-4754784 


8-657946 


583 


339339 


198155287 


24-1453929 


8-353905 


i 650 


422500 


274625000 


25-4950976 


8-662391 


534 


341056 


199176704 


24-1660919 


8-353678 


i 651 


423301 


275894451 


25-5147016 


8-666331 


535 


342225 


200201625 


24-1867732 


8-353447 


652 


425104 


277ir,78i)8 


25-5342907 


8-671266 


586 


343396 


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24-2074359 


8-353-209 


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425409 


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25-55.33647 


8 675697 


587 


344569 


202262003 


24 -2230 3-29 


8-3'72967 


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427716 


279726264 


25-5734237 


8-680124 


588 


345744 


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24 2487113 


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1 655 


429025 


281011375 


25-5929678 


8-684546 


5S9 


346921 


1 204336169 


24-2593222 


8-33246)5 


656 


43 336 


282.300416 


25-6124969 


8-688963 


590 


348100 


i 205379000 


24-2899156 


8-337206! i 657 


431649 


283593393 


25-6320112 


8693376 


591 


349281 


206425071 


24-3104916 


8-3919421 658 
8-396673 659 

8-401398 660 
8-406 118| 661 


43^964 


234890312 


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8-697784 


'>92 


350464 


1 207474688 


24-3310501 


434281 


286191179 


25-6709953 


8-702188 


593 


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1 208527857 


24-3515913 


435600 


28749(5000 


25 6901652 


8-706538 


594 


352836 


209584581 


24-3721152 


436921 


233804781 


25-7099203 
25-7293607 


8-710983 


595 


354025 


210644875 


24-3926218 


8-410333 662 


438244 


290117528 


8-715373 


596 


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24-4131112 


8-415542 663 


439569 


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8-719760 


597 


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24-4335834 


8-420245'' 664 


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24.4744765 


8-4296331 1 606 


443556 


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8-43432?!: 667 


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8-4183601 670 


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8-75:) 340 



APPENDIX. 



23 



No, 


Square. 


Cube. ! 


Sq. Root. 


CubeHoot. 


No. 


Square. 


Cube. 


Sq. Root. ICubeRoou 


67 J 


450241 


302111711 


25-90366771 8-754691 


7.38 


544644 


401947^72 


27-1661554 9-0368861 


672 


451584 


303464448 


25-9229628! 8-75903i 


739 


546121 


403583419 


27-1845544 


9 040965 


673 


452929 


304821217 


25-9422435 


8-763381 


740 


547600 


405224000 


27-2029410 


9-045042 


674 


454276 


306182024 


25-9615100 


8-7677191 74l| 


549081 


406869021 


27-2213152 


9-049114 


675 


455625 


307546875 


25-9807621 


8-772053! 


742 


550564 


408518488; 27-2390769 


9-053183 


676 


456971) 


308915776 


26 0000000 


8-776383! 


743 552049 


410172407 27-258u2b3 9-057248 


677 


458329 


310288733 


26-0192237 


8-78J708 


744 553556 


411830784 27-2763634 9061310 


678 


459684 


311665752 


26-0384331 


8-785030 


745 555925 


413493625 27-2946881 9-0b53oa 


679 


461041 


313046839 


26.0576284 


8-78i)347il 746 55b51u. 
8-793659 1 747 5.)8009 


415160936 27-3130006 


906yii22 


680 


46240e» 


314432(JO0 


26-0768096 


416832723 273313007 


9-073473 


681 


463761 


315821241 


26-0959767! 8-797968[i 74d| 5.:)9504| 


4l85j89y2 


27-3495887 


9 07/5^0 


6i2| 46.J124 


317214568 


261151297 


8-8022721 1 749 


561001 


420189749 


27-3678644 


9-0^1563 


683 


466489 


318611987 


26-1342687 


8-80'D572il 750 


56^500 


42187.0000 


27-3861279 


9-085b03 


684 


467856 


3-20013504 


26-1533937 


8-8108681I 75i 


564uOi 


4Z3564751 


27-40437y2 


9-08.)63j 


685 


469225 


321419125 


26-1725047 


8-815160 


752 


5b5504 


4:^5^59008 


27-4226184 


9-Oy3672 


686 


470596 


322828850 


26-1916017 


8-819447 


753 


5()7009 


426957777 


27-4408455 


9-oy7701 


(■)87 471%y 


3;^42427u3 


262106848 


8-823731 


754 


568516 


428661064 


27-45906j4 


9-101726 


688 


473344 


3;i566l672 


26-2297541 


8-828010 


755 


570025 


4^0568875 


27-4772633 


9-105748 


689 


474721 


3;i708276'J 


26-2488095 


8-832285 


756 


571536 


432081216 


27-4y54542 


9-109767 


690 


476100 


328509U00 


26-^678511 


8-836556 


757 


573049 


433798093 


27513633.> 


9-113/82 


691 


477481 


329939371 


26-2868789 


8-840823 


758 


574564 


4355 ly51^ 


27-5517yy8 


9-1177y3 


692 


478863 


531373888 


■/6-3U58929 


8845085 


759 


570081 


437245479 


27-5499546 


9-121801 


693 


480249 


332812557 


26-3248932 


8-84.. 344 


760 


577600 


438y76000 


27-5680975 


9- 125805 


»:94 


•181630 


354255384 


26-34387.)7 


8-853598 


761 


57^121 


44U711081 


27-5802284 


9-12y8o6 


695 


483J25 


33570^;i75 


26-3628527 


8-85 < 849 


762 


580644 


442450728 


27-6043475 


9- 133805 


696 


484416 


337153536 


26-3818119 


8-862095 


763 


5^2169 


444194947 


27 6224546 


y-l577j7 


697 


4858U9 


33«608873 


26-4007576 


8.866337 


764 


583096 


445y43744 


27-64u54ya 


9-141787 


698 


487204 


34006839^ 


26-41968^6 


8-870576 


765 


585z25 


447697 i25 


27 -6580354 


9-l'i5774 


699 


488601 


34153^099 


26-4386081 


8-874810 


76b 


580756 


44y455096 


27 67b7o5o 


9-i4y75o 


700 


490000 


343000000 


26-4575131 


8-879040 


767 


588^89 


451217663 


27-6947640 


9-155737 


701 


491401 


344472101 


^6-4764040 


8-883266 


768 


58y824 


452984832 


27-7l2812;> 


9-157714 


702 


492^04 


345948408 


26-4^52826 


8-887488 


769 


5yi36l 


454756609 


27-73084yi 


9-lulbO/ 


703 


494209 


347428927 


26-5141472 


8-891706 


770 


59^900 


450553000 


27-7488739 


9-165656 


704 


495616 


34 8913661 


26 5329i»83 


8-895920 


771 


5^4441 


458314011 


27-766o86o 


9 16902^ 


705 


4^7025 


35040,625 


26-5518361 


8-900130 


772 


5.>5-j84 


4000yy648 


27-7o4iobO 


9- 1/558 J 


706 


49^436 


351895816 


'Z6 5/06605 


8-904337 


773 


5975:i9 


461889917 


27-8028775 


y-i7/Di4 


707 


499849 


353393243 


:i6-5894716 


8-908539 


774 


59yo76 


463684824 


27-8z085o5 


9181UOI. 


708 


501264 


354894912 


mC) 0062694 


8-912/37 


775 


6006Z5 


465484375 


27-o38o2l8 


yio5455 


709 


5U2681 


356400829 


:.6 6^70539 


8-916931 


776 


602176 


407^885 /b 


27 -85077 bo 


9l8;;iO^ 


710 


504100 


357911000 


•^6tAb6zb^ 


8-921121 


777 


603729 


4byoy7433 


27-8747iy7 


y 1^5317 


711 


5J5521 


35942543! 


26 6645833 


8-925308 


778 


605284 


47091095*2 


27-8926514 


y-iu/^yo 


712 


506944 


360944128 


^6- 6833281 


8-929490 


779 


006841 


47V72913y 


2/-9io571o 


i>-;i0l22i> 


713 


508369 


362467097 


'4:6-702(j598 


8-933669 


78o 


60a4 


47^552000 


27-y^848ui 


i;-203ioi 


714 


50b7i)6 


353994344 


26-72o7784 


8-937843 


781 


60jy61 


47t 37^541 


27-y4b3772 


i;-;i«j:)Oi't 


715 


511225 


365525875 


;^^'" -7394839 


8-912014 


782 


6115i4 


4Vo21l768 


27-yb4:i;bzyl ^-^130^;^! 


716 


51^656 


367061696 


26-7581763 


8-946181 


783 


613089 


480048687 


27-i82i57^ 


■v-^lo-jOkj 


717 


514059 


368601813 


•^6-7768557 


8-950344 


784 


614056 


48183U304 


28-ooOOOoO 


y-»208/5 


718 


515524 


370l4C)23;i 


26-7 955220 


8-954503 


785 


6 16^25 


4oo/566z5 


28ol7o5l5 


v^Z-il'Ji 


719 


516%1 


37169495J 


26-8141754 


8-958658 


786 


6i77y6 


4e55c*7b5b 


28-0556i/l5 


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720 


518400 


373248U00 


26-8328157 


8-^jb2809 


78- 


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40744340 5 


28-u555<iw5 


J z-y'ioi'i) 


721 


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26-85 1443:J 


8-960U57 


780 


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722 


521284! 37t)3o704'- 


26 -870057 7 


1 8-i>71101 


1 789 


62^521 


491169069 


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y-2io43y 


723 


1 5a27:i9| 377933L)G7 


26-8880593 8-975241 


790 


6^4100 


4y5o3yooO 


28-lo6i)38b 


y-;i±i53o 


724 


521176 3795'o342'-: 


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6:^5601 


4^4913671 


28-1247ii^2 


a-;<i48^Jl 


725 


1 5^56ii5' 381078125 


26-9258240 8 983509 


792 


627264 


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28-l424i>^0 


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72b 


j 527076! 382657176 


;i6-9-k43872 8987657 


795 


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727 


528529 384240583 


25-9629375 8-99176^ 


i 7i>. 


b3u456 


50050618^ 


28-l7800ab 


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728 


52J984' 36582835, 


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i 79. 


632025 


50^:459875 


28-195/441 


y-^6.)7y7 


729 


531441! 3a7420489 


27-OOOOOuOl 9-000000 


! 79b 


633610 


5o43j8556 


2o-2l5472u 


y-zb7o8o 


730 


53:^900: 38i)0 17000 


27-0185122! 9-004113 


7i<. 


03j^oy 


506:;i6l575 


28-:65iloo4 


y.i71.ooy 


731 


1 534361: 391617891 


27-0370117; 9-0082^3 


! 798 


63000^ 


508iOu5i;^ 


z0-24doy3o 


;>-^i04o5 


73;^ 


! 535^24 3'.J2223168 


27-C551985' 9-01232'J 


799 


638401 


51O08ii3yy 


28-iibb5o8l 


y'^7i)3v^o 


733j 5^72^9 393^3^837 


27-0739727' 90164l31 


800 


640000 


512000000 


28-;^0'i271^ 


t> -203170 


734! 53^7;<6 3..544b9>*. 


27 -0 924344 9-020529 


; 801 


641601 


513y2240l 


28-5uly454 


y-ao/O"*-! 


735; 5411^25 3J7065375 


27.1108834' 9024621 


j 802 


643^04 


51584^608 


•28-3l9i.04O 


y-i:yU.07 


736' 5-il<i9..| 3ib&8b^otj 


271293199' 9-0.io7l5 


1 803 


644809 


517(81627 


'Za-o'6izb^(j 


y^i)4/b. 


737 1 54311.9' 400315553 


27-H7743.H 9-032802 


! 804 


646416 


5ly7i84b4 


z8-354ciy3C3 


y-2y80ii» 



u 



APPENDIX. 



No. 


Square, j 


Cube. 


Sq, Root. 


CubeRoot.] 


No. 1 


Square. 


Cute. 


Sq. Rrot- 


CubeRoot. 


805 


643025 


5216691251 28 3725219 


9-302477; 


872 


760384 


663;')4348 


29-5-296461 


9-553712 


806 


649636 


5236uC616i 28 3901391 


9-306323! 


873 


762129 


655333617 


29-5465734 


9-557363 


807 


651249 


5255579431 28-4077454 


9-310175 


874 


763376 


667627624 


29-5634910 


9-551011 


808 


652864 


5275141121 


28 4253408 


9-314019! 


875 


76562:. 


669921875 


29-5S03.)89 


9-564656 


809 


654181 


5294751291 


23-4429253 


9-317860! 


876 


767^76 


' 72221376 


29-5972972 


9-568298 


810 


656100 


531441000 


28-461)4989 


9-321697! 


877 


769129 


6745-25133 


29-6141853 


9-571li38 


811 


657721 


533411731 


23-4780617 


9-3255321 


878 


77'J884 


676836152 


29-6310648 


9-575574 


812 


659344 


535387328 


23-4956137 


9-3293631 


879 


772f)41 


679151439 


29-6479342 


9-579-08 


813 


660969 


537367797 


23-5131549 


9-3331921 


880 


774400 


68147-2O00 


29-6647939 


9-53 2H49 


814 


662596 


539353144 


23-5306852 


9-337017! 


08 1 


7761i.l 


683797341 


29-6816442 


9-:86468 


815 


664225 


511343375 


23-5482048 


9-340839i 


8^2 


777924 


686128968 


2l;.6984843 


9-590094 


816 


665856 


543338496 


23-5657137 


9-344657! 


883 


779689 


688465387 


29-7153159 


9-593717 


817 


6(i7489 


545333513 


28-5332119 


9-3-8473' 


8Si| 781456 


690807104 


29-7321375 


9-597337 


818 


669124 


547313432 


23-6006993 


9-352236' 


8l!5 7832:i5 


693154125 


29-7489496 


9-600955 


Sly 


670761 


549353259 


28-6181760 


9-356095 


886 


78499i. 


695506456 


29-7657521 


9-604570 


820 


672400 


551363000 


28-6356421 


9359902! 


837 


7867(59 


6978154103 


29-7825452 


9-60e432 


821 


674041 


553337661 


28-6530976 


9-363/ 05I 


833 


733544 


700227072 


29-79932S9 


9-61179! 


822 


675634 


555412248 


28-6705421 


9 30/5U5: 


839 


790321 


702595369 


29-8 16103 J 


9-61539- 


823 


677329 


557441767 


23-6379766 


9-37 1302 


890 


7^2100 


704969000 


29-8323678 


9-619002 


824 


678976 


559476224 


23-7054002 


9-3750961 


891 


793881 


707347971 


29-8496231 


9-622603 


825 


630625 


561515625 


28-7228132 


9 37338;! 


892 


795664 


709732288 


29-8663690 


9-626202 


826 


632276 


56355997(i 


28-74()2157 


y-3^-c67.') 


893 


79744'J 


712121957 


29-8331056 


9-62J7i^7 


827 


633J29 


565609283 


28-7576077 


9 -330460! 


894 


799236 


71451(5934 


29-8998328 


9-633391 


828 


635584 


567663552 


28-7749:^9 1 


9-39024-2' 


895 


801025 


71(3917375 


29-9165506 


9-636931 


82y 


687^41 


569722789 


28-7923601 


9-394021! 


896 


802816 


719323136 


21 .332591 


9-64056. 


830 


6889 '(- 


571737000 


28-8097206 


9-3 7796' 


897 


804609 


721734273 


29-9499533 


9-644154 


831 


6i>0561 


573356191 


28-8270706 


9-40156y| 


Sd6 


806404 


7241507'.?2 


29-9(566481 


9-647737 


832 


69-2^:4 


57593J368 


28-8444112 


9-4053391 


899 


8032 J I 


726572()9S 


29-9S33iJ37 


9-651317 


833 


693889 


578009537 


28-8617394 


9-4091051 


900 


810000 


729J00000 


30-0000000 


9-654894 


834 


695555 


580093704 


23-8790532 


9-412869! 


901 


811801 


731432701 


30-0166620 


9-6.58463 


835! 697225 


582182875 


2s-89636(i6 


9-41663j; 


902 


8136J4 


733o70808 


30-0333148 


y-662040 


836: 698890 


534277U56 


23-9136645 


9-4^0337' 


903 


815409 


735314327 


30-0499534 


y-665610 


837 700569 


586376253 


28-9309523 


9-4 24142 


904 


817216 


733753264 


30-0665-/23 


9-66yi76 


833 702244 


5i8480472 


28-94322J7 


9 4273i4 


905 


819025 


741217625 


30-0332179 


9-672740 


839 i 7u3y2l 


590539719 


23-9651967 


9-^31642 


906 


820836 


74.^677416 


30-0998339 


9-6763J2 


8401 705600 


592704000 


28-9827535 


9-435338 


907 


8^2649 


74614264-3 


30-1164407 


9-67-.860 


841 i 707281 


594323321 


29 0000000 


9-43yl31 


908 


824464 


748613312 


30- 1330383 


9-683417 


842 i 708.64 


596947633 


29-0172363 


9-44-2370 


909 


826231 


751089429 


30-1496269 


9-(;36970 


813| 710649 


599077107 


29-03446)23 


9-446607 


910 


828100 


753571000 


30-16(52063 


9-690521 


844 712336 


601211584 


29-051*.731 


9-450341 


911 


829921 


75(5053031 


30-1827765 


y-6y4069 


845 714025 


603351125 


29-0633837 


9-454072 


912 


831744 


753550528 


30-1993377 


y-6y76i5 


846 715716 


605495736 


29 0360791 


9-457800 


913 


833569 


761048497 


30-2158399 


y-701153 


847 


1 717409 


607(545423 


29-1032644 


9-461525 


91 i 


835396 


763551944 


30-232432J 


y-70469i. 


84^ 


719104 


61.9300192 


2912043Jf 


9-465247 


915 


837225 


76(5060375 


30-248J669 


9-708237 


84^ 


720801 


6 1 1960049 


29-1376046 


9-463966 


916 


839056 


763575296 


30ii654Jl9 


9-711772 


850' 722500 


614125000 


29-1547595 


9-4/^682 


91, 


840889 


771095213 


30-2820079 


9-7153.5 


85 li 724201 


6162' 5951 


29-1719043 


9-476396 


918 


842724 


1 7736-20632 


3 '2. '85 148 


9-713835 


852' 725904 


618470208 


29 1390390 


9-43OU.6 


919 


844561 


1 77(>151559 


303I50128 


9-7-22363 


853i 72760j 


62065;)477 


29-2L61637 


9-483314 


92 


846400 


773633000 


30-3315018 


9-725338 


854 


729316 


622335364 


29-2232; 84 


9-4a7518 


921 


848241 


781229961 


30-347i)318 


9-7-29411 


852 


7310-5 


6250^6375 


29-240333J 


9-49 1220 


9-22 


850084 


783777443 


30-364452-. 


9 73-2931 


856 


732736 


627222016 


29-2574777 


1 9-491^19 


9^3 


851929 


' 786330467 


30 3809151 


9-735448 


85; 


734449 


629422793 


i 29-27456^3: 9-4936 15 


9:^4 


853776! 783839024 


30-3^73633 


9-739963 


85S 


736164 


63162^712 


i 29 29 i 6370! 'J'oO-zSjS 


>.^25 


855625' 7914531-25 


30-4 138127 


9-743 17u 


85i 


737331 


6333c;y77J 


29-30370l8| 9-505998 


926 


857476; 79402277() 


3 -43 (2431 


9-746936 


860 73y6(H 


6261)56000 


! 29 3,57565! 9-509u85 j 'JZ, 


859329' 796597983 


39-4466747 


9-750493 


861 741321 


638^77331 


i 29-;^42:iOl5: y-51.j;.7w 


92j 


361134 


i 79917375- 


30-4630924 


9-753998 


862! 74304-1 


640503923 


1 -^935^8365 9 517u51 


929 


^63041 


1 801765039 


30-47.150 13 


9-757500 


863! 744769 


642735647 


i 29-3768616; y5;i0730 


i;3j 


864900: 804.)57<i00 


30-4..59014 


1 9-761000 


864| 746496 


644972544 


! 29-3933769 9-5244Uo 


931 


8667611 6^;695 -491 


30-5122926 


! 9-764497 


865 748^25 


6472146-.i5 


29-4103323 9-5280,9 


932 


863'';24 809557558 


30-528575^ 


9-7679-.(2 


8661 749y5t 


64946 139t 


I 29-4278779 \>bdi75j 


i.3;j 


87J4-.>. 312366237 


30-5150487 


9-77U84 


8671 75168a 


651714363 


I 29-4443637 9535417 


9b4 


8;2j5b; 81473v-5u4 


30-55 14lo6! 9-7749/4 


868j 753424 


653972032 


1 29-4618397 9-5^903:i 


^^35 


874-2-25! 8174U0375 


3-.;-57776;;7 9773462 


8691 755161 


656234909 


i 29-4738059 9-5i2744 


9-tj 


8760.,6: 8-2uu25:-:5u 


30-5.4il''l 9 7J1;J47 


870 75690L 


653503000 


5^9-4957624 9546403 


937 


8779691 82265(5953 


30-6lO455?i 9 7854^9 


8711 758641 


660776311 


1 2 J-5 1^7091, 9 05^059 


93^ 


8798441 8252^3672 


30-6267857i y-78j909 



APPENDIX. 



25 



No, 

939 
940 
941 
942 
943 
944 
945 
946 
947 
948 
949 
950 
951 
952 
953 
954 
955 
956 
957 
958 
959 
1-60 
961 
96-i 
963 
964 
965 
966 
967 
968 
969 



Square. Cube. Sq. Root 



881721 
883600 
885481 
83'' 364 
889249 
8yll36 
893025 
894916: 
8^6809' 
898704! 
90(;60r 
9025001 
904101! 
9063041 
908209 
9101161 
912025! 
9139361 
9158491 
9177641 
919681 I 
92iS0Oi 
923521: 
925144! 
927369! 
92U2u6: 
931225 
933156 
935089 
937024 
938961 



827936019: 
8305840001 
833237621! 
8353968881 
833561807! 
841232334! 
843908625! 
846590536^ 
849278123 
851971392, 
851670349 
857375000! 
860085 35 r 
862801408: 
865523177^ 
868250664 
8/0983875 
873722816 
87;)467493 
879217912 
881974079 
884736000 
887503681 
v^'J0277128 
a-.';i(i56347 
895841314 
898632125 
901428696 
10423 1063 
907039232 
909853209 



Cubelloot. 



3J-6431C69 
39-6594194 
30-6757233 
30-6920185 
30-7083051 
30-7245330 
30-7408523 
3i>- 757 1130 
30-7733651 
30-7896086 
30-8058436 
30-8221^700! 

30 8382879! 
30-8544972 
30-87069811 
30-8868904! 
30-9030743; 
30-9192497! 
3!)-93.54166i 
39-951575V 
30-9677-251! 
30-9^38668! 
31-00000001 
3101612481 
31-0322413! 

31 04834941 
31-0644491 
31-08054051 
31-09662361 
31-11269841 
31-12876481 



9 792386 
9-795361 
9-799334 

9-802304 
9-806271 
9-80J736 
9-813199 
9-816659 
9-82(Ul7 
9-823572 
9-827025 
9-83U476 
9-833924 
9-837369 
9-840813 
9-844254 
9-8476j2 
9-85] IvS 
9-854562 
9-857993 
9-861422 
9-864848 
9-868272 
9-871694 
9-875113 
9-878530 
9-881945 
9-885357 
9-883767 
9-892175 
9-895580 



No. 



970 
971 
972 
973 
974 
975 
976 
977 
978 
979 
980 
981 
982 
983 
984 
985 
986 
987 
288 
989 
990 
991 
992 
993 
994 
995 
996 
9..7 
998 
999 
1000 



Squi: 



Cube. 



940900 
942341 
944734 
946729 
918676 
950625 
952575 
954529 
956484 
958141, 
9604 00| 
96236 ij 
9643241 
9662891 
968256' 
970225! 
972196! 
9741691 
976144' 
9781211 
9801001 
982081 
984064 
986049 
988036 
990'025 
992016 
994i)09 
9960041 
99800 ij 
IGOOOOO! 



9126730C0 
915498611 
91d330()48 
921167317 
924010424 
926859375! 
929714176! 
93^574833 
935441352! 
93d3 13739 
911192000 
944076141 
946966163 
94986208' 
952763904 
955671625 
953535256 
961504803 
9644302"'2 
967361669 
970299000 
973242271 
976191488 
979146657 
982107784 
985071875 
988047936 
991026973 
994011992 
997002999 
1000000000 



Sq. Root. |CubeRc 



31-1448230J 
31-16087291 
31-1769145 
31-19294791 
31-20397311 
31 -22499001 
31-2409987! 
31-2569992 
31-2729915 
31-288.'757 
31-3041517 
31-3209195 
31-3368792 
31-3528308 
31-3687743 
31-3847097 
31-4006.369 
31-4165561 
31-4324673 
31-4433704' 
31-4642654! 
31-4801525 
31-4960315! 
31-5119025 
31-5277655 
31-5436206 
31-5594677 
31-5753068 
31-5911330 
316069613 
31-6227766 



9-8! 89 -BS* 
9-9023S3 
9-9057 -^2 
9-909173 
9-912571 
9-915962 
9-91935, 
9-92273 S 
9-926122 
9-9-29504 
9-9323 ! 1 
9-936261 
9-;;39633 
9-9430)9 
9-9163^0 
9-94y748 
9-953114 
9-956477 
9-959839 
9-963 19c. 
9-966555 
9-9699J9 
9-973262 
9-9766 12 
9-979960 
9-933305 
9-986619 
9-989990 
9-993.i29' 
9-996666, 

loooooooj 



The following rules are for finding the squares, cubes and roots, of 
numbers exceeding 1,000. 

To find the square of any number divisible without a remainder. 
Rule. — Divide the given number by such a number, from the forego- 
ing table, as will divide it without a remainder ; then the square of the 
quotient, multiplied by the square of the number found in the table, 
will give the answer. 

Example - What is the square of 2,000 ? 2,000, divided by 1,000, 
a number louna in the table, gives a quotient of 2, the square of which 
is 4, and the square of 1,000 is 1,000,000, therefore : 
4 X 1,000,000 = 4,000,000 : the Ans. 

Another example. — What is the square of 1,230? 1,230, being dl 
vided by 123, the quotient will be 10, the square of which is 100, and 
the square of 123 is 15,129, therefore : 

100 X 15,129-= 1,512,900: the Ans. 

To find the square of any number not divisible without a remainder. 
Rule. — Add together the squares of such two adjoining numbers, from 
the table, as shall together equal the given number, and multiply the 
sum by 2 ; then this product, less 1, will be the answer. 

Example. — What is the square of 1,487 ? The adjoining numbers, 
743 and 744. added^^together, equal the given number, 1,487, and the 
square of 743 = 552,049, the square of 744 = 553,536, and these 
added, = 1,105,585, iherefore : 

1,105,.585 X 2 = 2,211,170-^ I - 2,211,169: the Ans. 

To find the cube of any number u/ivisibie without a remainder. 
Rule. — Divide the given number by sucn a number, from the forego 



26 



APPENDIX. 



ing table, as will divide it without a reinainder ; then, the cube of tne 
quotient, multiplied by the cube of the number found in the table, will 
give the answer. 

Example.— VJ\\d.X is the cube of 2,700 ? 2,700, being divided by 900, 
the quotient is 3, the cube of which is 27, and the cube of 900 is 
729,000,000, therefore : 

27 X 729,000,000 -= 19,683,000,000: the Ans. 

To find the square or cube root of numbers higher than is found in the 
table. Rule. — Select, in the column of squares or cubes, as the case 
may require, that number which is nearest the given number ; then 
the answer, when decimals are not of importance, will be found di- 
rectly opposite in the column of numbers. 

Example. — What is the square-root of 87,620 ? In the column of 
squares, 87,616 is nearest to the given number ; therefore, 298, im- 
mediately opposite in the column of numbers, is the answer, laearly. 

Another example. — What is the cube-root of 110,591 ? In the co- 
lumn of cubes, 110,592 is found to be nearest to the given number; 
therefore, 48, the number opposite, is the answer, nearly. 

To find the cube-root more accurately. Rule. — Select, from the co- 
lumn of cubes, that number which is nearest the given number, and 
add twice the number so selected to the given number ; also, add twice 
the given number to the number selected from the table. Then, as 
the former product is to the latter, so is the root of the number selected 
to the root of the number given. 

Example. — What is the cube-root of 9,200? The nearest number 
in the column of cubes is 9,261, the root of which is 21, therefore : 
9261 9200 

2 2 



18522 18400 
9200 9261 



As 27,722 is to 27,661, so is 21 to 20-953 -^ the Ans. 

Thus, 2V661 X 21 = 580881, and this divided ty 27722 r= 20-953 + 

To find the square or cube root of a whole number with decimals. 
Rule. — Subtract the root of the whole number from the root of the next 
higher number, and multiply the remainder by the given decimal ; then 
the product, added to the root of the given whole number, will give the 
answer correctly to three places of decimals in the square root, and to 
seven in the cube root. 

Example. — W^hat is the square-root of 11*14? The square-root of 
11 is 3*3166, and the square-root of the next higher number, 12, is 
3'4641 ; the former from the latter, the remainder is 0*1475, and this by 
0-14 equals 0*02065e This added to 3*3166, the sum, 3*33725, is the 
square root of 11*14. 

To find the roots of decimals by the use of the table. Fule. — Seek for 
the gi\ en decimal in the column of numbers, and oppooite in the col- 
umns of roots will be found the answer, correct as to the figures, but re- 
quiring the decimal point to be shifted. The transposition of the deci- 
mal point is to be performed thus : For every place the decimal point is 
removed in the root, remove it in the number two places for the square 
root and three places for the cube root. 



APPENDIX. 27 

Examples, — By tlie table tlie square root of 86-0 is 9*2736, conse- 
quently, by the rule the sq lare root of 0*86 is 0-92'736. The square 
root of 9' is 3', hence the square root of 0*09 is 0'3. For the square 
root of 0*0657 we have 0*25632; found opposite No. 657. So, also, 
the square root of 0*000927 is 0*030446, found opposite No. 927. Aud 
the square root of 8*73 (whole number with decimals) is 2*9546, found 
opposite No. 873. The cube root of 0*8 is 0*928, found at No. 800 ; 
the cube root of 0*08 is 0*4308, found opposite No. 80, and the cube 
root of 0*008 is 0*2, as 2*0 is the cube root of 8*0. So also the cube 
root of 0*047 is 0*36088, found opposite No. 47. 



EULES FOR THE REDUCTIOi:^ OF DECIMALS. 

To reduce a fraction to its equivalent decimal. Rule. — Divide the 
numerator by the denominator, annexing cyphers as required. 

Example, — What is the decimal of a foot equivalent to 3 inches ? 
3 inches is j'^2 of a foot, therefore : 
!?_ , _ 12) 3-00 



•25 Ans. 
Another exam:)}e, — What is the equivalent decimal of |- of an inch? 



I ... 8) 7*000 



•875 An&. 

To reduce a compound fraction to its equivalent decimal. Mule, — In 
accordance with the preceding rule, reduce each fraction, commencing 
at the lowest, to the decimal of the next higher denomination, to which 
add the numerator of the next higher fraction, and reduce the sum to 
the decimal of the next higher denomination, and so proceed to the last ; 
and the final product will be the answer. 

Example. — What is the decimal of a foot equivalent to 5 inches, | 
and y^g- of an inch. 

The fractions in this case are, \ of an eighth, | of an inch, and -f^ of 
a foot, therefore : 

h 



^^ xu 




•5 

3- 


eighths. 


8) 3-5000 




•4375 
5- 


inches. 


r 12) 5-437500 




-453125 Ans, 



28 



APPENDIX. 



The process maybe condensed, thus; write the numerators of the 
given fractions, from the least to the greatest, under each other, and 
place each denominator to the left of its namerator, thus : 



i 



12 



1-0 



3-5000 



5-48'7500 



•453125 Ans. 

To reduce a decimal to its equivalent in terms of lower denominations. 
Rule. — Multiply the given decimal by the number of parts in the next 
less denomination, and point off from the product as many figures to 
the right hand, as there are in the given decimal ; then multiply the 
figures pointed ofi", by the number of parts in the next lower denomina- 
tion, and point off as before, and so proceed to the end ; then the seve- 
ral figures pointed off to the left will be the answer. 

Uxample. — What is the expression in inches of 0-390625 feet? 
Feet 0-390625 

12 inches in a foot. 



Inches 4-687500 

8 eighths in an inch. 



Eighths 5-5000 

2 sixteenths in an eighth. 

Sixteenth 1*0 

Ans., 4 inches, f and y^. 
Another example. — What is the expression, in fractions of an iach, 
of O'eS^S inches? 

Inches 0-6 8 7 5 

8 eighths in an inch. 



Eighths 5-5000 

2 sixteenths in an eighth. 



Sixteenth 1-0 



Ans.^ I and ^\ 




u 



V 



TABLE OF CIRCLES. 



(From Gregory's Mathematics.) 



From t'.iis table may be found by inspection the area or ciroumie' 
i'snce of a circle of any diameter, and the side of a square equal to the 
area of any given circle from 1 to 100 inches, feet, yards, miles, &o. 
If the given diameter is in inches, the area, circumference, &c., set 
opposite, will be inches ; if in feet, then feet, &c. 









Side of 








Side ol" 


Diam. 


Area. 


Circum. 


equal sq. 


Diam. 


Area. 


Circum. 


equal sq. 


•25 


•04908 


•78539 


•22155 


~~^ 


90-76257 


33-77212 


9-5-2693 


•5 


•19635 


1-57079 


•44311 


11- 


9503317 


34-55751 


9-74849 


•75 


•44178 


2-35619 


•66467 


•25 


99-40195 


35-34291 


9-97005 


!• 


•78539 


3-14159 


•886-22 


•5 


103-86890 


38-12331 


10-19160 


•25 


1-2-2718 


3-9-2699 


r 10778 


•75 


108-43403 


36-91371 


10-41316 


■5 


1-76714 


4-71238 


1-32934 


12- 


113-09733 


37-69911 


10-63472 


•75 


2-40528 


5-49773 


1-55983 


•25 


117^85881 


38-43451 


10856-27 


2- 


3-14159 


6-23318 


1-77245 


•5 


122-71846 


39-26990 


11-07783 


•25 


3-97607 


7-06858 


1-99401 


•75 


127-67628 


4005530 


11-29939 


•5 


4-90873 


7-85393 


2-21556 


13- 


132-73228 


40-84070 


11-52095 


•75 


5-93957 


8-63937 


2-43712 


•25 


137-88646 


41-6-2610 


11-74250 


3- 


7-06858 


9-42477 


2-65368 


-5 


143-13381 


42-41150 


11-95406 


•25 


8-29576 


10-21017 


2-88023 


-75 


148-48934 


43-19689 


1-2-18562 


•5 


9-62112 


10-99557 


3-10179 


14- 


153-93804 


43-98229 


12-40717 


•75 


11-04466 


11-78097 


3-3-2335 


-25 


159-48491 


44-76769 


12-62373 


4^ 


12-56637 


12-56637 


3-54490 


•5 


165-12996 


45-55309 


12-85029 


•25 


14-18625 


13-35176 


3-76646 


•75 


170-87318 


46-33349 


1307184 


•5 


15-90431 


14-13716 


3-98802 


15- 


176-71458 


47-12338 


13-29340 


•75 


17-7-2054 


14-92256 


4-20957 


•25 


182-65418 


47-90923 


13-51495 


5^ 


19-63195 


15-70796 


4-43113 


-5 


188-69190 


48-69468 


13-73551 


•25 


21-64753 


16-49336 


4-65269 


"75 


194-82783 


49-48003 


13-95307 


•5 


23-75839 


17-27875 


4-87424 


16- 


201-06192 


50-26518 


14-17963 


•75 


25-96722 


18-06415 


5-09580 


•25 


207-39420 


51-05088 


14-10118 


6- 


28 27433 


18-84955 


531736 


•5 


213-82464 


51-835-27 


14-62274 


•25 


30-67961 


19-63495 


5-53891 


•75 


220-35327 


52 62167 


14-84430 


•5 


33-18307 


20-42035 


5 76047 


17- 


226-98006 


53-40707 


15-05535 


•75 


35-78470 


21-20575 


5-93203 


•25 


233-70504 


54-19247 


15-28741 


7- 


33-48455 


21-99114 


620358 


•5 


240-52818 


54-97787 


15-50897 


25 


41-28249 


27-77654 


6-4-2514 


•75 


247-44950 


55 76326 


15-73052 


■f^. 


44-17864 


23-56194 


6-64670 


18^ 


264-46900 


55-54866 


15-9520.S 


•75 


47-17297 


24-34734 


6-86325 


•25 


266-58667 


57-33406 


16-17364 


8^ 


50-26548 


25-13274 


7-08981 


•5 


268 80252 


53-11946 


16 39519 


25 


53-45616 


2V91813 


7-31137 


•75 


276-11654 


58-90486 


15-61675 


•5 


5r)-7450l 


26-70353 


7-53292 


19^ 


283-52873 


59-69t)26 


16-83831 


•75 


6013204 


27-48893 


7-75448 


•25 


291-03910 


60-47565 


17-05936 


9- 


63-61725 


28-27433 


7-97604 


•5 


298-64765 


61-26105 


17-28142 


•25 


67-20063 


29-0:)97a 


8-19759 


•75 


305-35437 


62-04645 


17-50298 


•5 


70-83218 


29-84513 


8-41915 


20^ 


314-159-26 


62-83185 


17-72453 


•75 


74-66191 


30-63J52 


8-64071 


•25 


3-22-06233 


63-61725 


17-94509 


10^ 


78 53981 


31-41592 


8 86226 


-5 


330-06357 


64-40264 


18-15765 


•25 


82-5 1 589 


32-2)132 


9-03382 


-75 


338-16299 


65-18304 


18-38920 


•5 


86 59014 


32-986721 


9-30538 


21- i 


346-35059 


65 97344 


18-610761 



30 



APPENDIX. 









Side of 








Side of 


Diam. 


Area, 


Circum. 


equal sq. 


Diam. 


Area. 


Circum. 


equal sq. 


~21~25 


354-65635 


66-758841 


1883232^ 


^~~ 


1134-11494 


119-38052 


3367662 


•5 


363-05030 


67-54424 


19-05387 


•25 


1149-08660 


120-16591 


33-89817 


•75 


371-54241 


68-32964 


19-27543 


•5 


1164-15642 


120-95131 


34-] 1973 


22- 


380- 13-271 


69 11503 


19-49699, 


•75 


1179-32442 


121-73671 


34 34129 


•25 


388-82117 


69-90043 


19-71854 


39- 


1194-59060 


122-52211 


34 56-285 


•5 


3i)7-60782 


70-68583 


19-94010 


•25 


1209-95495 


123-30751 


3478440 


•75 


406-49263 


71-471-23 


20-16166 


-5 


12-25 41748 


124-09-290 


35 -(10596 


23- 


415-475(^2 


72-25663 


20-38321 i 


•75 


1240-97818 


124-87830 


35-22752 


•25 


424-55679 


73-04202 


20-60477: 


40- 


125-r63704 


125-66370 


35-44907 


•5 


433-73613 


73-82742 


20-82633 


•25 


1272-39411 


126-44910 


35 67063 


•75 


443-01365 


74-61282 


2104783 


•5 


1288-24933 


127-23450 


35-89219 


24^ 


452-38934 


75-398-22 


21-26944 


-75 


1304-20273 


128-01990 


36-11374 


•25 


461-8H320 


76-18362 


21-49100 


41- 


1320-25431 


123-805-29 


3" :^:;:)3.) 


•5 


471-43524 


76-96902 


21 71255 


•25 


1336-40406 


129-59069 


3o 55686 


•75 


481-10546 


77-75441 


21-93411 


•5 


1352-65198 


130-37609 


36 77841 


25- 


490-87385 


78-53981 


22-15567 


-75 


1368-99808 


131-16149 


36-99997 


•25 


500-74041 


79-32521 


22-37722 


42- 


1335-44236 


131-94689 


37-22153 


•5 


510-70515 


80-11061 


22-59878 


•25 


1401 98480 


132 73-2-28 


37--44308 


•75 


5-20-76S06 


80-89601 


22-82034 


•5 


1418-62543 


13351763 


37 66464 


26- 


530-92915 


81-68140 


23-04190 


■75 


1435-36423 


134-30308 


37-836-20 


•25 


541-18342 


82-46680 


23-26345 


43- 


1452-20120 


135-08348 


38 10775 


•5 


551-54586 


83-25-2-20 


23-48501: 


•25 


1469-13635 


135 87388 


38-32931 


•75 


562-00147 


84-03760 


23-70657: 


-5 


1486-16967 


136-659-28 


38 5598-/ 


27- 


572-55526 


84-82300 


23-9-2812: 


•75 


1503-30117 


137-44467 


33-77-242 


•25 


583-20722 


85-60839 


24 14968 


44- 


15-20-53084 


138-2.3007 


38-993j8 


•5 


593-95736 


86-39379 


•24-37124 


•25 


1537-85869 


139-01547 


39-21554 


•75 


604-80567 


87-17919 


24-59279: 


•5 


1556-23471 


139-80087 


39-43709 


28- 


615-75216 


87-96459 


24-81135 


•75 


1572-80890 


140-586-27 


39 65365 


•25 


6-26-79682 


8874999 


25-03591 


45- 


15 JO 43128 


141-37166 


39-83021 


•5 


637-93965 


89-53539 


25-25746 


•25 


1608 15182 


142-15706 


40-10176 


•75 


649-18066 


90-32078 


25-47'.;02: 


•5 


1625-97054 


142-94-246 


40-32332 


29- 


660 51985 


9M081^ 


25-70058 


•75 


1643 88744 


143-7-2786 


40 54488 


•25 


671-95721 


a 3915S 


25-92213 


46- 


1661-90251 


144 513-26 


40-76643 


•5 


683-49-275 


92-67698 


26-14369 


•25 


168a01575 


145-29866 


40-98799 


•75 


695-12646 


93-46-238 


26-36525, 


•5 


1698-22717 


146-08405 


41-20955 


30- 


706;85S34 


94-24777 


26-58680: 


•75 


1716-53677 


146-86945 


41-43110 


•25 


718 68840 


95 03317 


26-80836 


47^ 


1734-9445-4 


147-65485 


41-65-266 


•5 


730-61664 


95-81857 


27-02992 


•25 


1753-45048 


148-44025 


4187422 


•75 


74264305 


96-60397 


27-25147 


•5 


1772-05460 


149-2-2565 


42-09577 


31- 


75i^76763 


97-38937 


27-47303' 


•75 


1790-7.=.689 


150-01104 


42 31733 


•25 


766^9903^ 


98-17477 


27-69459: 


48^ 


1809 557.^6 


150-79644 


42-53889 


•5 


779-31132 


98 96016 


27-91614: 


•25 


18-28-45601 


151-58184 


42-76044 


•75 


• 791-73043 


99-74556 


28-13770' 


•5 


1847-45-282 


152-36724 


42-98-200 


32- 


804-24771 


100-53096 


23-^i:.9-26' 


•75 


1866-54782 


153-15264 


43-20356 


•25 


816-86317 


101-31636 


•2'v53081 


49- 


1885-74099 


153-93804 


43 4-2511 


•5 


8-29-57681 


102-10176 


28-:' -0237 


•25 


1905-83233 


154-72343 


43-64667 


•75 


842-38861 


102-88715 


29 U2393 


•5 


19-24-42184 


155-50883 


43-86823 


33- 


855-29859 


10367255 


29-21548 


•75 


1943-90954 


156-29423 


44-08978 


•25 


868-30675 


104-45795 


29-46704 


50- 


1963-49540 


157-07963 


44-31134 


•5 


881-41308 


105-24335 


29-68860 


•25 


1983-17944 


157-96503 


44-53290 


•75 


894-61759 


106-0-2875 


29-91015 


•5 


2002-96166 


153-65042 


44-75445 


34^ 


907-92027 


106-81415 


30-13171 


■75 


2022-84205 


159-43582 


44-97601 


25 


921-32113 


107-59954 


30-35327 


5h 


2042-82062 


160-22122 


45-19757 


•5 


934-82016 


108-38494 


30-57482 


•25 


2062-89736 


161-0(662 


45-41912 


•75 


94841736 


109-17034 


30-79638 


•5 


2083-072-27 


161-79202 


45-64068 


35- 


962-11275 


109-95574 


31-01794 


-75 


2103-34536 


162-57741 


45-852-24 


•25 


975-90630 


110-74114 


31-23949 


52- 


2123-71063 


163-36-281 


46-05:130 


•5 


989-79803 


111-52653 


31-46105 


•25 


2144-18607 


164-14821 


46-30535 


•75 


1003-78794 


112-31193 


31-68261 


•5 


2164-75368 


164-93361 


46 5-2691 


36- 


1017-87001 


11309733 


31-90416 


•75 


2185-41947 


165-71901 


46-74847 


•25 


1032-06-227 


113-88-273 


32-12572 


53- 


2-206-18344 


166-50441 


46-97(K)2 


•5 


1046-34670 


114-66813 


32-34728 


•25 


2227-04557 


167-28980 


47-19158 


•75 


1060-72930 


115-45353 


32-56883 


•5 


2-248-00589 


168075-20 


47-41314 


37^ 


1075-21008 


1 16-238 J2 


32-79039 


•75 


2-269-06433 


168-86060 


47-63469 


•25 


1089-78903 


117-02432 


33-01195 


54- 


2-290-22104 


169-64600 


47-8562.-) 


•5 


1104-46616 


117-80972 


33-23350 


•25 


2311-475-^8 


170-43140 


48-0778'! 


•75 


1119-24 14" 


1 18-59572 


33-4551j6 


•5 


2332-8-2889 


171-21679 


48-29930 



APPENDIX. 



■^■■""' 






Side of 


1 


1 




Side of 


Diani. 


Area. 


Circinr.. 


equal sq. 


Diam. 


Area. | 


Circum. 


equal sq. 


54-75 


•2351 -280(8 


172-00219 


48-52092 


71-5 


~40i505m' 
4043 27833 


2-24 623^7 


63-365-22 


55- 


2375-82. '44 


172 -7^759 


4874248 


■75 


2-25-40 27 


63-58678 


•25 


23;;7- 47698 


173-5 7 2.)9 


48-96403 


72- 


4071-50407 


2-26- 19467 


63-80833 


•5 


24]'J-2-22r)9 


174-3583.) 


4918559 


•25 


4099-82750 


226-;;8i)-;:6 


64-02989 


•75 


2 141 -06657 


175-1437J 


49-40715 


-5 


4128 24909 


227 76546 


64-25145 


56- 


24;.3 00864 


175-92918 


49-62870 


•75 


4156-763861 


228 55.86 


64-47300 


•25 


2185-;)4887 


176-71458 


49-85026 


73^ 


4185 386811 


2^i;-3^626 


64-69456 


•5 


2507-187-28 


177-49998 


5007182 


•25 


4214-10293 


23i)12i6 


6491612 


•75 


2520-4-^337 


178-28538 


50-29337 


•5 


4242-91722! 


230-9t.7< 6 


65 13767 


57^ 


2551-75S63 


179-07078 


5i)51493 


-75 


4271-82969: 


231 69-245 


65-35023 


•25 


2574-19156 


179-85617 


50-73649 


74- 


4300-84034' 


232-47785 


65 58079 


•5 


25'J6-7-2-267 


180-64157 


50 95304 


-25 


43-29-949 16' 


233 263-i5 


65 80234 


•75 


2619-35196 


181-426.7 


51-17960 


•5 


4359-15615' 


'^3 4-048. 5 


66-02390 


58^ 


2642(;7942 


182 21237 


51-40116 


-75 


4388-46 i>2' 


234 834(5 


66-24546 


•25 


26; )4 i^(>5n5 


182-99777 


51-6-2-271 


75- 


4417-86466 


235-61944 


66-46701 


•5 


2687-82886 


183-78317 


51844-27 


•25 


4447 36618: 


236-40484 


66-68857 


•75 


2710-85084 


184-56856 


52-06583 


■5 


4476 -V 65 ^8: 


237-19C24 


66-91043 


59- 


2733-97100 


185-353.6 


52-28738 


•75 


4506-663741 


237-97-64 


67-13168 


•25 


275718933 


186 13936 


52-50894 


76^ 


4536-45079' 


238-761(4 


67-35324 


•5 


2780-50584 


186-92476 


52-73050 


•25 


4566 35400 


239-54613 


67-57180 


•75 


2803 92053 


187-71016 


52-95205 


•5 


45.6 34640 


240-33183 


67-79635 


60- 


2827-43338 


188-49555 


5317364 


•75 


462643696 


241-11723 


68-01791 


•25 


2851-04442 


189-28095 


53 39517 


77^ 


4656 6-2571! 


241-90-263 


68-23047 


■5 


2874-75362 


190-06635 


53 61672 


•25 


463691262! 


242-68803 


68-46102 


•75 


2898-56100 


190-85175 


53-83828 


•5 


4717-297711 


243-47343 


68-68258 


61 


2922-46656 


191-63715 


54-05984 


•75 


4747-78098 


244-25,382 


68-90414 


•25 


2946-47029 


192-42255 


54-28139 


78^ 


4778-36242, 


245 044-22 


69-1-2570 


•5 


2^^70-572-20 


193-20794 


54-50205 


-25 


4809-04204 


245-82962 


69 31725 


•75 


2994-77228 


198-99334 


54-7-2451 


•5 


4839-81983' 


246-61502 


69-56881 


B2- 


3019-07054 


194-77874 


54 94606 


•75 


4870-79579 i 


247-40042 


69-79037 


•25 


3043 46697 


195-56414 


55-16762 


79- 


4901-66993 


248-18581 


70-01192 


•5 


3a67-;:6i57 


190-34J54 


55-3 J9 18 


•25 


4932-74225 


248-97121 


70-23318 


•75 


3092 55435 


197-13493 


55 61073 


•5 


4963-91274 


249-75661 


70-4.5504 


63- 


3117-24531 


197 92033 


55-83229 


•75 


4995-18140, 


250-34^01 


70-67659 


•25 


3142 03444 


198-70573 


58-05335 


80- 


5026-548-24 


251 32741 


70-89815 


•5 


3166-92174 


199-49113 


56-27540 


•25 


5058-01325 


25211281 


71-11971 


•75 


■ 3191-90722 


200-27653 


56-496j6 


•5 


5089-57644! 


252-8ii8-20 


71-341-26 


64- 


3216-99087 


201-06bt2 


56 7135 i 


-75 


5121-23781: 


253 63360 


71-56-282 


•25 


3242 17270 


201-84732 


56-94007 


81- 


5152-99735' 


254-46000 


71-78438 


•5 


3-267-45270 


20263272 


57-16163 


•25 


5184-85506 


255-25440 


72-00593 


•"5 


3292 83088 


203-41812 


57-383191 


•5 


5216-81095 


256-03./80 


72-2-2749 


65- 


3318-30724 


204-20352 


57-60475' 


•75 


5218-86501, 


256-82579 


72-44905 


•25 


3313-88176 


204-98892 


57-82631' 


82- 


5281-017-25 


257-61059 


■ 2-67060 


•5 


3369-55447 


205-77431 


58 04786 


•25 


5313 26766 


258-3. 509 


72-80216 


•75 


339532534 


206-55971 


53-26042 


•5 


5345 61624' 


259-18139 


73 11372 


66- 


342119439 


207-34511 


53 49097 


•75 


5378-06301 


259-96679 


73-33527 


25 


344716162 


203- 13051 


58-71253 


83^ 


54 10-60^ 94i 


260-75219 


73-55683 


•5 


347322702 


208-91591 


58-93400 


•25 


544325105 


261-53758 


7377839 


•75 


3199 39060 


209^70130 


50-15564 


•5 


.5475-i:9234: 


262-3^-298 


73-99994 


67^ 


35-25-65235 


210 48r.70 


59-37720 


•75 


5508-83180 


263-10838 


74-22150 


25 


3552-01228 


211-27210 


5950876 


84^ 


5541-76944, 


263-8.; 378 


74-44306 


•5 


3578-47033 


212^05750 


59-82031 


•25 


5574-80525' 


264-67918 


74-66461 


•75 


3t)05 02655 


212-84-290 


6004187 


•5 


5607-93023 


265-46457 


74-88617 


68- 


3631-68110 


213-62-30 


60-26343 


•75 


5641-17130 


266-24997 


75']0773 


•25 


3658-43373 


214-41369 


60-43498 


85^ 


5674 50173 


267-03537 


75-32y2j 


•5 


3685-28453 


215-19909 


60-70654 


• -25 


5707-93023J 


267-82077 


75-55084 


•75 


3712-23350 


215-98149 


60-92810 


•5 


5711-456:;2 


268-60617 


75 77240 


69^ 


3 "^39 -28065 


216 76089 


61 -14-065 


•75 


5775-08178 


269-39157 


75-99395 


•25 


376G-42597 


217-555-29 


61-37121 


86^ 


5308-80-481 


270-17696 


76-2155! 


•5 


37 9;r 66947 


218 34068 


61-50-277 


•25 


5842-6-26021 


270-96236 


76 437. 7 


•75 


3321-01115 


219-1-2608 


61-81432 


•5 


5876-54540 


271-74776 


76-65 S6 2 


70 


3848-45100 


219-91 1-48 


62-03538 


•75 


5010-56296 


272 53316 


76-88'i| ^ 


25 


3875-98902 


2-20-69683 


62-25744 


87^ 


5944 67369 


273 31-56 


:7 lom 


•5 


3:i03-625-22 


2^1-48-2-28 


62-47809 


•25 


5978-89260' 


274-103.)5 


77-323^0 


•75 


3931-35959 


2-22 -^i! .7 68 


62-7* '055 


tr 


6013-201C8 


274-88J35 


77-51 18r. 


71 


3959-19214 


223-05307 


62-92211 


'75 


6047 61494 


275 67475 


777664] 
77'98:y6 


•25 


3987-12-286 


223S;-io47 


03 14366 


88^ 


6082-12337 


276-46015 



33 



APPENDIX. 









Side of 








Side ol 


Diam. 


Area. 


Circum. 


equal sq. 


Diam. 


Area. 


Circuii. 


equal sq. 


~88~25 


6110-72993 


277-24555 


73-20952 


94-25 


6976-74097 


2r 6-09510 


83-52688 


•5 


6 151 •43476 


278 03094 


78-43103 


•5 


70 IS -80 194 


296-88050 


83-74344 


■75 


6186-23772 


278-81634 


7665263 


-75 


7050-96109 


297-66590 


83-97U0J 


89- 


6221-13885 


279-60174 


78-87419 


95- 


708V21342 


2J8-45i30 


84-19155 


•25 


6256-13815 


28033714 


79-09575 


-25 


7125-57992 


299-23u70 


84-4131 ! 


•5 


6291-23563 


231-17254 


79-31730 


-5 


716302759 


300-0-2209 


84-63467 


•75 


6326-43129 


281-95794 


79-53336 


•75 


720057944 


300-80749 


84-85622 


90- 


6361-72512 


282-74333 


79-760^2 


96- 


7233-22947 


3u 1-59239 


85-07778 


•25 


6397-11712 


233-52873 


79-98 19:< 


•25 


7275-97767 


302-37829 


85-29934 


•5 


6432-60730 


284-31413 


80 20353 


-5 


7313-82404 


3>)3- 16369 


85-52089 


•75 


646319566 


285-0^953 


80-42509 


•75 


7351-76359 


303-94908 


85-74245 


9r 


6503-83219 


285-8S493 


80-64659 


97- 


7339-81131 


304-73448 


85-96401 


.25 


653J-66639 


286-67032 


80-86820 


•25 


7427-95221 


305-5 1983 


86-185-.6' 


•5 


6575-54977 


237-45572 


81-03976 


-5 


7466- 19 129 


306-30523 


86-40712 


•75 


6611-53J82 


288-24112 


81-31132 


•75 


7504-52853 


307-091)68 


86-6-2368 


92^ 


6847-61005 


289-02652 


81-53^37 


98^ 


7542-96396 


307-87603 


86-85023 


•25 


6683-78745 


289-81192 


81-75443 


•25 


7581-49755 


308-66147 


87-07179 


•5 


6720-0630:3 


2.iO-oJ7:]z 


81-975J9 


•5 


7620-12933 


309-44637 


87-29335 


•75 


6756-43678 


291-33271 


82-10754 


•75 


7653-H5927 


310-23227 


87-51490 


93- 


6792-90871 


292-16811 


8241910 


99- 


76i)7-68739 


311-01767 


87-73646 


•25 


6829^47831 


292-95351 


82-04066 


-25 


7736-61369 


311-80307 


87-95802 


•5 


6866-14709 


2:)3-73S9J 


82-862^1 


•5 


7775-63316 


312-58346 


88-17957 


•75 


6902-91354 


2-J4-52431 


83-08377 


■75 


7814-76031 


31337336 


88-40113 


94^ 


6939-77817 


295 30970 


S3 Hi I') J 3 


1100- 


7353-98163 


314-15926 


88^622a9 



The following rules are f:)r extending the use of the above table. 

To find the area, circumference, or side of equal square, of a circle 
having a diameter of more than 100 inches, feet, Sj'C. Rule. — Divide 
the given diameter by a number that will give a quotient equal to some 
one of the diameters in the table : then the circumference or side of 
equal square, opposite that diameter, multiplied by that divisor, or, the 
area opposite that diameter, multiplied by the square of the aforesaid 
divisor, will give the answer. 

Exam-pie. — What is the circumference of a circle whose diameter is 
228 feet ? 228, divided by 3, gives 76, a diameter of the table, the cir- 
cumference of which is 238-761, therefore : 
238-761 
3 



716-283 feet. Ans. 
Another example. — What is the area of a circle having a diameter 
of 150 inches ? 150, divided by 10, gives 15, one of the diameters in 
the table, the area of which is 176-71458, therefore : 
176-71458 

100= 10 X 10 



17,671-45800 inches. Ans. 
To find the area, circumference, or side of equal square, of a circle 
having an intermediate diameter to those in the table. Rule. — Multiply 
tlie given diameter by a number that will give a product equal to some 
one of the diameters in the table ; then the circumference or side of 
equal square opposite that diameter, divided by that multiplier, or, the 
area opposite that diameter divi led by th( square of the aforesaid mul 
tiplier, will give the answer. 



APPENDIX. 



Example. — What is the circumference of a circle whose diameter ia 
6j^, or 6-125 inches ? 6-125, multiplied hy 2, gives 12-25, one of thf 
diameters of the tahh;, whose circuniference is 38-484, therefore : 
2)38-484 



19-242 inches. Ans. 
Another example. — What is the area of a circle, the diameter oi 
which is 3-2 feet '? 3-2, multiplied hy 5, gives 16, and the area of 16 
is 201-0619, therefore : 

5 X 5 — 25)201-0619(8-0424 + feet. Ans. 
200 

106 
100 

61 

50 

119 

100 

19 

Note. — The diameter of a circle, multiplied hy 3-14159, will give 
Its circumference ; the square of the diameter, multiplied by -78539, 
will give its area ; and the diameter, multiplied by '88622, will give 
the side of a square equal to the area of the circle. 



TABLE SHOWING THE CAPACITY OF WELLS, CISTERNS, AC. 



The gallon of the State of New York, by an act passed April 11, 1851, is required to conform 



to the standard gallon of the United States g^ 
inches. In conformity with this standard th 

One foot in depth of a cistern of 

3 feet diameter will contain 

3i " 

u 

u 

u 

u 



4 


(( 


4i 
5 


u 
u 


6 




1 




8 


u 


9 


u 





u 


2 


u 



ernnient. This standard gallon contains 231 cubi* 
following table has been computed. 



52-872 gallons, 

71-965 " 

93-995 " 

118-963 " 

146-868 " 

177-710 « 

211-490 " 

248-207 " 

287-861 « 

375-982 " 

475-852 " 

587-472 " 

845-959 •* 



Vote — To reduce cubic feet to gall )ns, multiply by 7*48, 



/^ Cr ^ 



TABLE OF POLYGONS 

(From Gregory's Mathematics.) 



. 1> 


Names. 


Multipliers for 


Radius of cir- 


Factors for 


^■7^ 




areas. 


cuin. circle. 


sides. 


3 


Trigon 


0-4330127 


0-5773503 


J -732051 


4 


Tetragon, or Square 


1-0000000 


0-7071068 


1-414214 


5 


Pentagon - 


1-7204774 


0-8506508 


1-175570 


6 


HexajTon 


2-5980762 


1-0000000 


1-000000 


7 


Heptagon - 


3-63391-24 


1-15238-24 


0-867767 


8 


Octagon 


4-8284271 


1-3065628 


0-765367 


9 


Nonagon - 


6-1818242 


1-4619022 


0-684040 


10 


Decagon 


7-6942088 


1-6180340 


0-6181)34 


11 


Undecagon 


9-3656399 


1-7747324 


0-563465 


12 


Dodecagon - 


11-1961524 


1-9318517 


0-517638 



To find the area of any regular 'polygon, whose sides do not exceed 
twelve. Riu'e. — Multiply the square of a side of the given polygon by 
the number in the column termed Multipliers for areas, standing op 
posite the name of the given polygon, and the product will be the an- 
swer. Example. — What is the area of a regular heptagon, whose 
sides measure each 2 feet ? 

3-6339124 

4 = 2X2 



14-53564-96: Ans. 
To find the radius of a circle which will circumscrihe any regular 
'polygon given, whose sides do not exceed twelve. Rul?. — Multiply a 
side of the given polygon by the number in the column termed Radius 
of circumscrihing circle, standing opposite the name of the given poly- 
gon, and the product will give the answer. Example. — What is the 
radius of a circle which will -circumscribe a regular pentagon, whose 
sides measure each 10 feet ? 

•8506508 
10 



8-5065080: Ans. 
To find the side of any regular 'polygon that may he inscribed within 
a given circle. Rule. — Multiply the radius of the given circle by the 
number in the column termed Factors for sides, standing opposite the 
name of the given polygon, and the product will be the answer. Ex- 
ample. — What is ttie side of a regular octagon that may be inscribeij 
within a circle, whose radius is 5 feet ? 
•765367 
5 



3-826835: Ans. 



WTJGHT CF IMATKRIALS 



Woods. 


lbs. in a 
cubic foot 


Metals. 


ff)s. in a 
cuJnc fool 


Apple, - 


. 49 


Wire-drawn brass. 


- 534 


Ash, 


40 


Cast brass, 


506 


Beach, - - 


. 40 


Sheet-copper, 


- 540 


Birch, 


45 


Pure cast gold, - 


- 1210 


Box, 


- 60 


Bar-iron, 


475 to 487 


Cedar, 


28 


Cast iron, - 


450 to 475 


Virginian red cedar. 


- 40 


Milled lead, - 


- 713 


Cherry, 


38 


Cast lead. 


709 


Sweet chestnut. 


- 36 


Pewter, 


- 453 


Horse-chestnut, 


- - 34 


Pure platina, 


1345 


Cork, 


- 15 


Pure cast silver, 


- 654 


I 'ypress, 


28 


Steel, 


486 to 490 


Ebony, - 


- 83 


Tin, 


- 456 


Elder, 


43 


Zinc, 


439 


Elm, - - 


- 34 


Stone, Earths, SfC. 




Fir, (white spruce,) 


29 


Brick, Phila. stretchers, 105 


Hickory, 


- 52 


North river common hard 


Lance-wood, 


59 


brick. 


- 107 


Larch, - 


- 31 


Do. salmon 


brick, 100 


Larch, (whitewood,) 


22 


Brickvvork, about 


95 


Lignunm-vitsB, - 


- 83 


Cast Roman cement 


- 100 


Logwood, 


57 


Do. and sand in equal parts, 1 1 3 


Rt. Domingo mahogany, - 45 


V halk. 


144 to 166 


Honduras, or bay maho 


gany, 35 


C.iy, - - - 


- 11& 


Maple, 


47 


Potter's clay, 


112 to 130 


White oak, 


43 to 53 


Common earth, 


95 to 124 


Canadian oak, 


54 


Flint, - 


- 163 


Red oak. 


- 47 


Plate-glass, 


172 


Live oak. 


7fi 


Crown-glass, - 


- 157 


White pine, 


23 to 30 


Granite, 


158 to 187 


Yellow pine. 


34 to 44 


Quincy granite. 


. 166 


Pitch pine. 


46 to 58 


Gravel, 


109 


Poplar, 


25 


Grindstone, - 


- 134 


Sycamore, 


- 36 


Gvpsum, (Plaster-stone,) 142 


Walnut, 


40 

t 


Unslaked lime. - 
)5 


52 



30 



APPENDIX. 



Ihs. in a 
cuhic foot. 

Limestone, - - 118 to 198 


Common blue stone, 


\hs. in a 
cubic fool 

16C 


Marble, - - 16] to 177 
New mortar, - - - 107 


Silver-gray flagging, 
Stonework, about, 


- 185 
120 


Dry mortar, 


90 


Common plain tiles. 


. 115 


Mortar with hair, (Plaster- 




Sundries. 




ing,) - - . . 
Do. dry. 


105 

86 


Atmospheric air. 
Yellow beeswax, - 


- 0-075 
. 60 


Do. do. including lath 




Birch-charcoal, - 


34 


and nails, from 7 to 1 1 




Oak-charcoal, 


- 21 


lbs. per superficial foot. 
Crystallized quartz, 
Pure quartz-sand. 


165 
171 


Pine-charcoal, 
Solid gunpowder, - 
Shaken gunpowder. 


17 
- 109 

58 


Clean and coarse sand. 


100 


Honev, 


- 90 


Welsh slate, - 


180 


Milk."' 


64 


Paving stone. 


151 


Pitch, - 


- 71 


Pumice stone, 


56 


Sea-water, 


04 


Nyack brown stone, - 


148 


Rain-water, .. 


- 62-5 


Connecticut brown stone, 


170 


Snow, 


8 


i'arrytown blue stone, - 


171 


Woo:'-ashes, 


53 



THE END. 



New York, July, 1874. 

JOHN WILEY &, SON'S 

LIST OF PUBLICATIONS, 

15 ASTOR. PLACE, 

Under the Mercantile Library and Trade Svolerooms. 



AGRICULTURE. 



DOWNING. FRUITS AND FRUIT-TREES OF AMERICA; or che 

Culture, Propagation, and Management in the Gard(?n and 
Orchard, of Fruit-trees generally, with descriptions of ah the 
finest varieties of Fruit, Native and Foreign, cultivated in this 
country. By A. J. Downing. Second revision and con oction, 
with large additions. By Chas. Downing. 1 vol. 8\o, over 
1100 pages, with several hundred outline engravings. Price, 
with Supplement for 1872 $5 00 

•'As a work of roference it has no equal in this conntry, and deserves a place in 
the Library of every Pomolo^ist in America." — Marshall P. Wilder. 

•• ENCYCLOPEDIA OF FRUITS; or, Fruits and Fruit- 
Trees of America. Part 1. — Apples. With an Appendix 
containing many new varieties, and brought down to 1872. 
By Clias. Downing. With numerous outline engravings. 8vo, 
full cloth $2 50 

** ENCYCLOPEDIA OF FRUITS; or, Fruits and Fruit- 
Trees of America. Part 2. — Ciieiuues, Grapes, Peaches, 
Peaks, &C. With an Appendix containing many new varie- 
ties, and brought down to 1872. By Chas. Downing. With 
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A FEW FROM MANY TESTIMONIALS. 

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COMBINATION, AND APPLICATION OF CALCA- 
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